On the difference between variational and unitary coupled cluster theories

Harsha, Gaurav; Shiozaki, Toru; Scuseria, Gustavo E.

doi:10.1063/1.5011033

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:

^[1];

^[2];

^[3]

Rice Univ., Houston, TX (United States). Dept. of Physics and Astronomy
Northwestern Univ., Evanston, IL (United States). Dept. of Chemistry
Rice Univ., Houston, TX (United States). Dept. of Physics and Astronomy, and Dept. of Chemistry

Publication Date:: 2018-01-28

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.  doi:10.1063/1.5011033.

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                    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

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                    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.

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@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}

}

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

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Works referencing / citing this record:

Variational Quantum Simulation for Quantum Chemistry
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Similar Records

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

Abstract

Citation Formats

Figures / Tables:

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

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

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

Structure of the exact wave function journal, August 2000

Benchmark variational coupled cluster doubles results journal, November 2000

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

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

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

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

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

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

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

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

Can the Eigenstates of a Many-Body Hamiltonian Be Represented Exactly Using a General Two-Body Cluster Expansion? journal, September 2003

Validity of many-body approximation methods for a solvable model journal, October 1966

Exactly-Solvable Models Derived from a Generalized Gaudin Algebra text, January 2004

Variational Quantum Simulation for Quantum Chemistry journal, January 2019

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

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Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian
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Benchmark studies of variational, unitary and extended coupled cluster methods
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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

Structure of the exact wave function
journal, August 2000

Benchmark variational coupled cluster doubles results
journal, November 2000

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

An Introduction to Coupled Cluster Theory for Computational Chemists
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Stability conditions and nuclear rotations in the Hartree-Fock theory
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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
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Alternative ansätze in single reference coupled‐cluster theory. III. A critical analysis of different methods
journal, July 1995

Alternative single-reference coupled cluster approaches for multireference problems: The simpler, the better
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Extended coupled-cluster method. IV. An excitation energy functional and applications to the Lipkin model
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An extension of the coupled cluster formalism to excited states
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Use of a size-consistent energy functional in many electron theory for closed shells
journal, January 1983

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

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

The Lipkin pseudospin model in the variational exp S formalism
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Can the Eigenstates of a Many-Body Hamiltonian Be Represented Exactly Using a General Two-Body Cluster Expansion?
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Validity of many-body approximation methods for a solvable model
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Exactly-Solvable Models Derived from a Generalized Gaudin Algebra
text, January 2004

Variational Quantum Simulation for Quantum Chemistry
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