Anharmonic vibrational many-body theory for molecules and extended systems

Qin, Xiuyi

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https://hdl.handle.net/2142/117732 Copy

Description

Title

Anharmonic vibrational many-body theory for molecules and extended systems

Author(s)

Qin, Xiuyi

Issue Date

2022-10-14

Director of Research (if dissertation) or Advisor (if thesis)

Hirata, So

Doctoral Committee Chair(s)

Hirata, So

Committee Member(s)

• Ceperley, David
• Makri, Nancy
• Pogorelov, Taras

Department of Study

Chemistry

Discipline

Chemistry

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Anharmonicity
• Many-body theory
• Quantum chemistry
• Vibrational analysis

Abstract

Anharmonic vibrational many-body theories have been developed and applied for molecules and extended systems. The second-order Green's function method for anharmonic crystals has been applied to an infinite, periodic chain of polyethylene taking into account up to quartic force constants. The frequency-independent approximation to the Dyson self-energy gives rise to numerous divergent resonances, which are fortuitous. Instead, solving the Dyson equation self-consistently with a frequency-dependent self-energy resists divergences from resonances or zero-frequency acoustic vibrations. The calculated anharmonic phonon dispersion, which nonetheless displays many true resonances, and anharmonic phonon density of states furnish hitherto unknown details that explain smaller features of the observed vibrational spectra. Thermodynamic functions of an ideal molecular gas due to its anharmonic vibrations are evaluated in a wide range of temperatures ($T$) by the vibrational full-configuration-interaction (FCI) method using a quartic force field and a finite number ($N$) of harmonic-oscillator basis functions along each normal mode. The thermodynamic functions considered are the grand potential ($\Omega$), internal energy ($U$), and entropy ($S$). They are compared with those obtained from the Bose--Einstein theory with or without truncation of the harmonic-oscillator basis functions after quantum number $N-1$. The comparison reveals that the finite-basis-set errors in $\Omega$ and $U$ are, respectively, $k_\text{B}T \ln (k_\text{B}T/N\hbar \omega)$ and $k_\text{B}T$ per mode in the high-$T$ limit, obscuring anharmonic effects when $k_\text{B}T > \hbar\omega$, where $\omega$ is the lowest mode frequency. The benchmark data for several low-order perturbation corrections to $\Omega$, $U$, and $S$ are also obtained as the numerical derivatives of their FCI values with respect to dimensionless perturbation strength, and the domain of $T$ and $N$ in which these data are reliable (for the $N \to \infty$ limits) is discussed. The finite-temperature many-body perturbation theory for anharmonic vibrations is comprehensively and systematically developed. The vibrational Rayleigh-Schr\"odinger style algebraic recursions for thermodynamic functions in the grand canonical ensemble such as the grand potential, internal energy, and entropy, are introduced. The perturbation corrections to the energies entering the recursions are evaluated by the Hirschfelder–Certain degenerate perturbation theory, which incorporates the resonances caused by degenerate states. The recursions serve as a mathematical basis for two distinct, but equivalent derivations. One is the algebraic (finite-temperature Born-Huang) derivation relying on the finite-temperature Born-Huang rules and thermal factorization for mathematical reduction. The other is the second-quantized derivation formulated by the quantum-field-theoretical techniques of normal ordering of ladder operators and Wick's theorem at finite temperature. The Feynman diagrams at finite temperature are introduced as a mnemonic tool to substitute the rigorous second-quantized derivations by the direct translations of diagrams to reduced analytical formulas. The perturbation corrections to energies enter the recursions in the form of a nondiagonal matrix, which gives rise to special diagrams, renormalization diagrams, where the resolvent lines are shifted from their original places. It is distinct from the well-known anomalous diagrams whose resolvent lines are completely removed. As a numerical benchmark method, the general-order algorithm is developed by directly implementing the recursions in a full configuration interaction (FCI) program. This recursion method is compared with $\lambda$-variation method, which is the numerical differentiation of the finite-temperature FCI with respect to perturbation strength, for perturbation corrections to thermodynamic functions up to high orders and it shows convergence towards the FCI limit at low and high temperatures. On the other hand, the derived reduced analytical formulas are verified by the exact agreement with these two numerical benchmark methods at low temperatures as the former is basis-set-free while the latter two suffers from basis-set errors at high temperature.

Graduation Semester

2022-12

Type of Resource

Thesis

Copyright and License Information

Copyright, Xiuyi Qin, 2022

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