Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm

Qin, Xiuyi; Hirata, So

doi:10.1063/5.0164326

Title: Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm

Abstract

A unified theory is presented for finite-temperature many-body perturbation expansions of the anharmonic vibrational contributions to thermodynamic functions, i.e., the free energy, internal energy, and entropy. The theory is diagrammatically size-consistent at any order, as ensured by the linked-diagram theorem proved in this study, and thus applicable to molecular gases and solids on an equal footing. It is also a basis-set-free formalism, just like its underlying Bose-Einstein theory, capable of summing anharmonic effects over an infinite number of states analytically. It is formulated by the Rayleigh-Schrödinger-style recursions, generating sum-over-states formulas for the perturbation series, which unambiguously converges at the finite-temperature vibrational full-configuration-interaction limits. Two strategies are introduced to reducing these sum-over-states formulas into compact sum-over-modes analytical formulas. One is a purely algebraic method that factorizes each many-mode thermal average into a product of one-mode thermal averages, which are then evaluated by the thermal Born-Huang rules. Canonical forms of these rules are proposed, dramatically expediting the reduction process. The other is finite-temperature normal-ordered second quantization, which is fully developed in this study, including a proof of thermal Wick's theorem and the derivation of a normal-ordered vibrational Hamiltonian at finite temperature. The latter naturally defines a finite-temperature extension of size-extensive vibrational self-consistentmore »« less

Authors:

Qin, Xiuyi ^[1];

^[1]

University of Illinois at Urbana-Champaign

Publication Date:: Sun Jan 01 00:00:00 EST 2023

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

Sponsoring Org.:: USDOE Office of Science (SC), Basic Energy Sciences (BES). Chemical Sciences, Geosciences & Biosciences Division (CSGB)

OSTI Identifier:: 1993171

Grant/Contract Number:: SC0006028

Resource Type:: Accepted Manuscript

Journal Name:: The Journal of Chemical Physics

Additional Journal Information:: Journal Volume: 159; Journal Issue: 8; Journal ID: ISSN 0021--9606

Country of Publication:: United States

Language:: English

Subject:: 74 ATOMIC AND MOLECULAR PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Anharmonic vibrations; perturbation theory; thermodynamics; Feynman diagrams; quantum field theory

Citation Formats


                    Qin, Xiuyi, and Hirata, So. Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm.  United States: N. p., 2023. 
Web.  doi:10.1063/5.0164326.

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                    Qin, Xiuyi, & Hirata, So. Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm.  United States.  https://doi.org/10.1063/5.0164326

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                    Qin, Xiuyi, and Hirata, So. Sun .  
"Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm".  United States.  https://doi.org/10.1063/5.0164326.  https://www.osti.gov/servlets/purl/1993171.

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@article{osti_1993171,

  title        = {Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm},

  author       = {Qin, Xiuyi and Hirata, So},

  abstractNote = {A unified theory is presented for finite-temperature many-body perturbation expansions of the anharmonic vibrational contributions to thermodynamic functions, i.e., the free energy, internal energy, and entropy. The theory is diagrammatically size-consistent at any order, as ensured by the linked-diagram theorem proved in this study, and thus applicable to molecular gases and solids on an equal footing. It is also a basis-set-free formalism, just like its underlying Bose-Einstein theory, capable of summing anharmonic effects over an infinite number of states analytically. It is formulated by the Rayleigh-Schrödinger-style recursions, generating sum-over-states formulas for the perturbation series, which unambiguously converges at the finite-temperature vibrational full-configuration-interaction limits. Two strategies are introduced to reducing these sum-over-states formulas into compact sum-over-modes analytical formulas. One is a purely algebraic method that factorizes each many-mode thermal average into a product of one-mode thermal averages, which are then evaluated by the thermal Born-Huang rules. Canonical forms of these rules are proposed, dramatically expediting the reduction process. The other is finite-temperature normal-ordered second quantization, which is fully developed in this study, including a proof of thermal Wick's theorem and the derivation of a normal-ordered vibrational Hamiltonian at finite temperature. The latter naturally defines a finite-temperature extension of size-extensive vibrational self-consistent field theory. These reduced formulas can be represented graphically as Feynman diagrams with resolvent lines, which include anomalous and renormalization diagrams. Two order-by-order and one general-order algorithms of computing these perturbation corrections are implemented and applied up to the eighth order. The results show no signs of Kohn-Luttinger-type nonconvergence.},

  doi          = {10.1063/5.0164326},

  journal      = {The Journal of Chemical Physics},

  number       = 8,

  volume       = 159,

  place        = {United States},

  year         = {Sun Jan 01 00:00:00 EST 2023},

  month        = {Sun Jan 01 00:00:00 EST 2023}

}

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Olsen, Jeppe; Christiansen, Ove; Koch, Henrik
The Journal of Chemical Physics, Vol. 105, Issue 12
DOI: 10.1063/1.472352

High-order coupled-cluster calculations through connected octuple excitations
journal, April 2000

Hirata, So; Bartlett, Rodney J.
Chemical Physics Letters, Vol. 321, Issue 3-4
DOI: 10.1016/s0009-2614(00)00387-0

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Title: Finite-temperature many-body perturbation theory for anharmonic vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm

Abstract

Citation Formats

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Anharmonic contributions to vibrational thermodynamic properties of solids journal, September 1961

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The lattice dynamics of an anharmonic crystal journal, October 1963

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journal, November 2012

Second-Order Many-Body Perturbation Study on Thermal Expansion of Solid Carbon Dioxide
journal, December 2014

Ice Ih anomalies: Thermal contraction, anomalous volume isotope effect, and pressure-induced amorphization
journal, May 2016

Role of Low-Energy Phonons in Thermal Conduction
journal, August 1954

Theory of Thermal Conductivity of Anharmonic Crystals
journal, May 1973

Theory of thermal conductivity of anharmonic crystals: Nondiagonal Peierls contribution
journal, November 1974

Temperature pulses in dielectric solids
journal, October 1968

Helmholtz Free Energy of an Anharmonic Crystal toO(λ4)
journal, June 1971

Fermi resonance in solid CO ₂ under pressure
journal, February 2013

Response to “Comment on ‘Fermi resonance in solid CO ₂ under pressure’” [J. Chem. Phys. 140, 177101 (2014)]
journal, May 2014

Anharmonic Phonon Dispersion in Polyethylene
journal, November 2020

Thermodynamic limit and size-consistent design
journal, June 2011

Extensivity of Energy and Electronic and Vibrational Structure Methods for Crystals
journal, May 2012

On the mutual exclusion of variationality and size consistency
journal, January 2014

The use of a model in anharmonic lattice dynamics
journal, January 1958

Theory of the Self-Consistent Harmonic Approximation with Application to Solid Neon
journal, July 1966

Properties of Crystalline Argon and Neon in the Self-Consistent Phonon Approximation
journal, January 1968

First-Order Dyson Coordinates and Geometry
journal, May 2013

Helmholtz free energy of an anharmonic crystal toO(λ4). II
journal, October 1974

Anharmonic contributions to vibrational thermodynamic properties of solids
journal, September 1961

Anharmonic contributions to vibrational thermodynamic properties of solids
journal, September 1961

Anharmonic contributions to vibrational thermodynamic properties of solids
journal, May 1963

The lattice dynamics of an anharmonic crystal
journal, October 1963

Anharmonic crystals
journal, January 1968

Helmholtz free energy of an anharmonic crystal to O(λ4). III. Equation of state for the Lennard-Jones solid
journal, January 1985

Helmholtz free energy of an anharmonic crystal to O(λ4). IV. Thermodynamic properties of Kr and Xe for the Lennard-Jones, Morse, and Rydberg potentials
journal, August 1985

Anharmonic lattice vibrations in solid N₂
journal, July 1978

Anharmonic lattice vibrations in solid N₂
journal, December 1978

Un développement du potentiel de gibbs d'un système quantique composé d'un grand nombre de particules
journal, June 1958

Ground-State Energy of a Many-Fermion System
journal, April 1960

Ground-State Energy of a Many-Fermion System. II
journal, June 1960

Formulation de la me´canique statistique en termes de nombres d'occupation (I)
journal, May 1961

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journal, October 1955

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The anharmonic force field of ethylene, C ₂ H ₄ , by means of accurate ab initio calculations
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General method for removing resonance singularities in quantum mechanical perturbation theory
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Anharmonicity in the ring stretching modes of diborane
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An efficient approach for calculating vibrational wave functions and zero-point vibrational corrections to molecular properties of polyatomic molecules
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Vibrational zero-point energies and thermodynamic functions beyond the harmonic approximation
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Anharmonic vibrational properties by a fully automated second-order perturbative approach
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Simple(r) algebraic equation for transition moments of fundamental transitions in vibrational second-order perturbation theory
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Treatment of Fermi resonance effects on transition moments in vibrational perturbation theory
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A fully automated implementation of VPT2 Infrared intensities
journal, August 2010

Simple perturbation theory estimates of equilibrium constants from force fields
journal, January 1991

Coherent state semiclassical initial value representation for the Boltzmann operator in thermal correlation functions
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The self-consistent-field approach to polyatomic vibrations
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Self‐consistent field energies and wavefunctions for coupled oscillators
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Excited vibrational states of polyatomic molecules: the semiclassical self-consistent field approach
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Investigations of self-consistent field, scf ci and virtual stateconfiguration interaction vibrational energies for a model three-mode system
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Mo/ller–Plesset perturbation theory applied to vibrational problems
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Vibrational coupled cluster theory
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First-principles theories for anharmonic lattice vibrations
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The Linear Response Approximation and Its Lowest Order Corrections: An Influence Functional Approach
journal, April 1999

Size-extensive vibrational self-consistent field methods with anharmonic geometry corrections
journal, June 2012

Second-order many-body perturbation expansions of vibrational Dyson self-energies
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Higher-order diagrammatic vibrational coupled-cluster theory
journal, October 2015

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journal, February 2018

Modeling physical systems by effective harmonic oscillators: The optimized quadratic approximation
journal, February 1995

Time‐dependent coupled cluster method: A new approach to the calculation of molecular absorption spectra
journal, June 1988

Coupled cluster description of anharmonic molecular vibrations. Application to O3 and SO2
journal, January 1994

Calculation of vibrational energy of molecule using coupled cluster linear response theory in bosonic representation: Convergence studies
journal, October 2008

Diagrammatic theories of anharmonic molecular vibrations
journal, December 2014

Normal-ordered second-quantized Hamiltonian for molecular vibrations
journal, November 2014

Finite-temperature many-body perturbation theory in the canonical ensemble
journal, February 2020

Finite-temperature many-body perturbation theory in the grand canonical ensemble
journal, July 2020

Degenerate RS perturbation theory
journal, February 1974

Normal ordered exponential approach to thermal properties and time-correlation functions: general theory and simple examples
journal, October 2021

One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms
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