skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: On the difference between variational and unitary coupled cluster theories

Abstract

There have been assertions in the literature that the variational and unitary forms of coupled cluster theory lead to the same energy functional. Numerical evidence from previous authors was inconsistent with this claim, yet the small energy differences found between the two methods and the relatively large number of variational parameters precluded an unequivocal conclusion. Using the Lipkin Hamiltonian, we here present conclusive numerical evidence that the two theories yield different energies. The ambiguities arising from the size of the cluster parameter space are absent in the Lipkin model, particularly when truncating to double excitations. We show that in the symmetry adapted basis under strong correlation, the differences between the variational and unitary models are large, whereas they yield quite similar energies in the weakly correlated regime previously explored. We also provide a qualitative argument rationalizing why these two models cannot be the same. Additionally, we study a generalized non-unitary and non-hermitian variant that contains excitation, de-excitation, and mixed operators with different amplitudes and show that it works best when compared to the traditional, variational, unitary, and extended forms of coupled cluster doubles theories

Authors:
ORCiD logo [1]; ORCiD logo [2]; ORCiD logo [3]
  1. Rice Univ., Houston, TX (United States). Dept. of Physics and Astronomy
  2. Northwestern Univ., Evanston, IL (United States). Dept. of Chemistry
  3. Rice Univ., Houston, TX (United States). Dept. of Physics and Astronomy, and Dept. of Chemistry
Publication Date:
Research Org.:
Rice Univ., Houston, TX (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES)
OSTI Identifier:
1512944
Alternate Identifier(s):
OSTI ID: 1417782
Grant/Contract Number:  
SC0001474; FG02-09ER16053
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 148; Journal Issue: 4; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Harsha, Gaurav, Shiozaki, Toru, and Scuseria, Gustavo E. On the difference between variational and unitary coupled cluster theories. United States: N. p., 2018. Web. https://doi.org/10.1063/1.5011033.
Harsha, Gaurav, Shiozaki, Toru, & Scuseria, Gustavo E. On the difference between variational and unitary coupled cluster theories. United States. https://doi.org/10.1063/1.5011033
Harsha, Gaurav, Shiozaki, Toru, and Scuseria, Gustavo E. Sun . "On the difference between variational and unitary coupled cluster theories". United States. https://doi.org/10.1063/1.5011033. https://www.osti.gov/servlets/purl/1512944.
@article{osti_1512944,
title = {On the difference between variational and unitary coupled cluster theories},
author = {Harsha, Gaurav and Shiozaki, Toru and Scuseria, Gustavo E.},
abstractNote = {There have been assertions in the literature that the variational and unitary forms of coupled cluster theory lead to the same energy functional. Numerical evidence from previous authors was inconsistent with this claim, yet the small energy differences found between the two methods and the relatively large number of variational parameters precluded an unequivocal conclusion. Using the Lipkin Hamiltonian, we here present conclusive numerical evidence that the two theories yield different energies. The ambiguities arising from the size of the cluster parameter space are absent in the Lipkin model, particularly when truncating to double excitations. We show that in the symmetry adapted basis under strong correlation, the differences between the variational and unitary models are large, whereas they yield quite similar energies in the weakly correlated regime previously explored. We also provide a qualitative argument rationalizing why these two models cannot be the same. Additionally, we study a generalized non-unitary and non-hermitian variant that contains excitation, de-excitation, and mixed operators with different amplitudes and show that it works best when compared to the traditional, variational, unitary, and extended forms of coupled cluster doubles theories},
doi = {10.1063/1.5011033},
journal = {Journal of Chemical Physics},
number = 4,
volume = 148,
place = {United States},
year = {2018},
month = {1}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 4 works
Citation information provided by
Web of Science

Figures / Tables:

FIG. 1 FIG. 1: Typical configurations with even and odd parity for the N = 8 site Lipkin model. (a) Even parity. (b) Odd parity.

Save / Share:

Works referenced in this record:

Coupled-cluster theory in quantum chemistry
journal, February 2007


Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian
journal, February 2017

  • Wahlen-Strothman, Jacob M.; Henderson, Thomas M.; Hermes, Matthew R.
  • The Journal of Chemical Physics, Vol. 146, Issue 5
  • DOI: 10.1063/1.4974989

Benchmark studies of variational, unitary and extended coupled cluster methods
journal, December 2010

  • Cooper, Bridgette; Knowles, Peter J.
  • The Journal of Chemical Physics, Vol. 133, Issue 23
  • DOI: 10.1063/1.3520564

Extended coupled-cluster method. I. Generalized coherent bosonization as a mapping of quantum theory into classical Hamiltonian mechanics
journal, September 1987


New perspectives on unitary coupled-cluster theory
journal, January 2006

  • Taube, Andrew G.; Bartlett, Rodney J.
  • International Journal of Quantum Chemistry, Vol. 106, Issue 15
  • DOI: 10.1002/qua.21198

Error analysis and improvements of coupled-cluster theory
journal, January 1991


The Lipkin–Meshkov–Glick model as a particular limit of the <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mrow><mml:mi mathvariant="italic">SU</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> Richardson–Gaudin integrable models
journal, May 2013


Exactness of wave functions from two-body exponential transformations in many-body quantum theory
journal, January 2004


Structure of the exact wave function
journal, August 2000

  • Nakatsuji, Hiroshi
  • The Journal of Chemical Physics, Vol. 113, Issue 8
  • DOI: 10.1063/1.1287275

Benchmark variational coupled cluster doubles results
journal, November 2000

  • Van Voorhis, Troy; Head-Gordon, Martin
  • The Journal of Chemical Physics, Vol. 113, Issue 20
  • DOI: 10.1063/1.1319643

Exactly-solvable models derived from a generalized Gaudin algebra
journal, February 2005


An Introduction to Coupled Cluster Theory for Computational Chemists
book, January 2000

  • Crawford, T. Daniel; Schaefer, Henry F.; Lipkowitz, Kenny B.
  • Reviews in Computational Chemistry: Lipkowitz/Reviews
  • DOI: 10.1002/9780470125915.ch2

Validity of many-body approximation methods for a solvable model
journal, February 1965


Stability conditions and nuclear rotations in the Hartree-Fock theory
journal, November 1960


Validity of many-body approximation methods for a solvable model
journal, February 1965


On certain correspondences among various coupled-cluster theories for closed-shell systems
journal, March 1982

  • Pal, Sourav; Prasad, M. Durga; Mukherjee, Debashis
  • Pramana, Vol. 18, Issue 3
  • DOI: 10.1007/bf02847816

Alternative coupled-cluster ansätze II. The unitary coupled-cluster method
journal, February 1989


Alternative ansätze in single reference coupled‐cluster theory. III. A critical analysis of different methods
journal, July 1995

  • Szalay, Péter G.; Nooijen, Marcel; Bartlett, Rodney J.
  • The Journal of Chemical Physics, Vol. 103, Issue 1
  • DOI: 10.1063/1.469641

Alternative single-reference coupled cluster approaches for multireference problems: The simpler, the better
journal, June 2011

  • Evangelista, Francesco A.
  • The Journal of Chemical Physics, Vol. 134, Issue 22
  • DOI: 10.1063/1.3598471

Extended coupled-cluster method. IV. An excitation energy functional and applications to the Lipkin model
journal, October 1989


An extension of the coupled cluster formalism to excited states
journal, January 1981


Use of a size-consistent energy functional in many electron theory for closed shells
journal, January 1983

  • Pal, Sourav; Durga Prasad, M.; Mukherjee, Debashis
  • Theoretica Chimica Acta, Vol. 62, Issue 6
  • DOI: 10.1007/bf00557928

Brueckner based generalized coupled cluster theory: Implicit inclusion of higher excitation effects
journal, September 2000

  • Nooijen, Marcel; Lotrich, Victor
  • The Journal of Chemical Physics, Vol. 113, Issue 11
  • DOI: 10.1063/1.1288912

Validity of many-body approximation methods for a solvable model
journal, February 1965


The Lipkin pseudospin model in the variational exp S formalism
journal, October 1983


    Works referencing / citing this record:

    Variational Quantum Simulation for Quantum Chemistry
    journal, January 2019

    • Li, Yifan; Hu, Jiaqi; Zhang, Xiao‐Ming
    • Advanced Theory and Simulations, Vol. 2, Issue 4
    • DOI: 10.1002/adts.201800182

    An adaptive variational algorithm for exact molecular simulations on a quantum computer
    journal, July 2019


    Unitary coupled-cluster based self-consistent polarization propagator theory: A third-order formulation and pilot applications
    journal, June 2018

    • Liu, Junzi; Asthana, Ayush; Cheng, Lan
    • The Journal of Chemical Physics, Vol. 148, Issue 24
    • DOI: 10.1063/1.5030344

    Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms
    journal, July 2019

    • Bauman, Nicholas P.; Bylaska, Eric J.; Krishnamoorthy, Sriram
    • The Journal of Chemical Physics, Vol. 151, Issue 1
    • DOI: 10.1063/1.5094643

    Exact and approximate symmetry projectors for the electronic structure problem on a quantum computer
    journal, October 2019

    • Yen, Tzu-Ching; Lang, Robert A.; Izmaylov, Artur F.
    • The Journal of Chemical Physics, Vol. 151, Issue 16
    • DOI: 10.1063/1.5110682

    Exact parameterization of fermionic wave functions via unitary coupled cluster theory
    journal, December 2019

    • Evangelista, Francesco A.; Chan, Garnet Kin-Lic; Scuseria, Gustavo E.
    • The Journal of Chemical Physics, Vol. 151, Issue 24
    • DOI: 10.1063/1.5133059

    Quantum algorithms for electronic structure calculations: Particle-hole Hamiltonian and optimized wave-function expansions
    journal, August 2018

    • Barkoutsos, Panagiotis Kl.; Gonthier, Jerome F.; Sokolov, Igor
    • Physical Review A, Vol. 98, Issue 2
    • DOI: 10.1103/physreva.98.022322

      Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.