Theoretical treatment of anharmonicity of vibrational modes of single‐walled carbon nanotubes
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Abstract We report a computational study, using the “moments method” [Y. Gao and M. Daw, Modell. Simul. Mater. Sci. Eng. 23 045002 (2015)], of the anharmonicity of the vibrational modes of single‐walled carbon nanotubes. We find that modes with displacements largely within the wall are more anharmonic than modes with dominantly radial character, except for a set of modes that are related to the radial breathing mode that are the most anharmonic of all. We also find that periodicity of the calculation along the tube length does not strongly affect the anharmonicity of the modes but that the tubes with larger diameter show more anharmonicity. Comparison is made with available experiments and other calculations.Low-frequency molecular vibrations at far-infrared frequencies are thermally excited at room temperature. As a consequence, thermal fluctuations are not limited to the immediate vicinity of local minima on the potential energy surface, and anharmonic properties cannot be ignored. The latter is particularly relevant in molecules with multiple conformations, such as proteins and other biomolecules. However, existing theoretical and computational frameworks for the analysis of molecular vibrations have so far been limited by harmonic or quasi-harmonic approximations, which are ill-suited to describe anharmonic low-frequency vibrations. Here, we introduce a fully anharmonic analysis of molecular vibrations based on a time correlation formalism that eliminates the need for harmonic or quasi-harmonic approximations. We use molecular dynamics simulations of a small protein to demonstrate that this new approach, in contrast to harmonic and quasi-harmonic normal modes, correctly identifies the collective degrees of freedom associated with molecular vibrations at any given frequency. This allows us to unambiguously characterize the anharmonic character of low-frequency vibrations in the far-infrared spectrum.
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A numerical analysis has been made of anharmonicity effects for vibrational eigenmodes in a model of vitreous silica constructed by molecular dynamics. It is shown that in macroscopic systems all eigenmodes are generally harmonic. Strong anharmonicity is found for the bare localized modes in the low-frequency region around the boson peak. This anharmonicity is related to the rotational motion of coupled ${\mathrm{SiO}}_{4}$ tetrahedra and differs in character from the anharmonicity of the high-frequency localized eigenmodes which are mainly related to symmetric and asymmetric stretching of ${\mathrm{SiO}}_{4}$ tetrahedra.
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By a suitable choice of coordinates, the computational effort required for calculations of anharmonic vibrational spectra can be reduced significantly. By using suitable localized-mode coordinates obtained from an orthogonal transformation of the conventionally used normal-mode coordinates, anharmonic couplings between modes can be significantly reduced. However, such a transformation introduces harmonic couplings between the localized modes. To elucidate the role of these harmonic couplings, we consider the vibrational self-consistent field (VSCF)/vibrational configuration interaction (VCI) calculations for both few-mode model systems and for ethene as a molecular test case. We show that large harmonic couplings can result in significant errors in localized-mode L-VSCF/L-VCI calculations and study the convergence with respect to the size of the VCI excitation space. To further elucidate the errors introduced by harmonic couplings, we discuss the connection between L-VSCF/L-VCI and vibrational exciton models. With the help of our results, we propose an algorithm for the localization of normal modes in suitable subsets that are chosen to strictly limit the errors introduced by the harmonic couplings while still leading to maximally localized modes.
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Strain-induced coupling constants for the anharmonicity of Raman-active lattice and certain molecular modes of vibration in 1,2,4,5-tetrabromobenzene (TBB) crystals have been determined using piezomodulated Raman spectroscopy. These constants, which are directly related to the first anharmonic term in the potential energy expansion for lattice dynamical calculations, are a quantitative measurement of the modal anharmonicities in the TBB molecular crystal. Application of uniaxial stress in the experiments permits the anisotropy of the anharmonicity to be determined as well as its magnitude. The TBB lattice modes are significantly coupled by the induced strains and the effects of coupling were observed to be dependent on the direction and symmetry of the strains. The molecular modes investigated were, by comparison, less coupled by the acoustic phonons and generally exhibited less anharmonic response with increasing frequency.
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This paper reveals that vibrational excitations in amorphous solids induce particle rearrangements and cause transitions to different states, which do not concur in crystals. These results suggest a rather complex structure of the energy landscape in amorphous solids.
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The vibrations of atoms within a crystalline lattice can give insight into properties of materials. In particular, vibrational anharmonicity is responsible for many critical phenomena, such as thermal effects, specifically in the low frequency range. While vibrational anharmonicity can be measured spectroscopically, quantum-mechanical simulations traditionally can not take anharmonicity into account. In this work, two recently developed methods, the VCI and VSCF models, are used to calculate the explicit anharmonicity in molecular crystals.
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Thermodynamic properties of selected small and medium size molecules were calculated using harmonic and anharmonic vibrational frequencies. Harmonic vibrational frequencies were obtained by normal mode analysis, whereas anharmonic ones were calculated using the vibrational self-consistent field (VSCF) method. The calculated and available experimental thermodynamic data for zero point energy, enthalpy, entropy, and heat capacity are compared. It is found that the anharmonicity and coupling of molecular vibrations can play a significant role in predicting accurate thermodynamic quantities. Limitations of the current VSCF method for low frequency modes have been partially removed by following normal mode displacements in internal, rather than Cartesian, coordinates.
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