Probing anharmonic phonons by quantum correlators - A path integral approach

At a Glance

Metadata	Details
Publication Date	2021-06-10
Journal	The Journal of Chemical Physics
Authors	T. Morresi, L. Paulatto, R. Vuilleumier, M. Casula
Institutions	Centre National de la Recherche Scientifique, École Normale Supérieure - PSL
Citations	14
Analysis	Full AI Review Included

Executive Summary

This research introduces a novel and highly efficient computational scheme for determining vibrational properties (phonons) in strongly anharmonic quantum systems using Path Integral Molecular Dynamics (PIMD).

Core Innovation: The method utilizes zero-time, Kubo-transformed correlation functions (force-force and displacement-displacement) as quantum phonon estimators.
Efficiency Breakthrough: Solving the phonon equations via Generalized Eigenvalue Equations (GEV), rather than standard eigenvalue problems, yields an overall efficiency gain of greater than 10 times (faster convergence and smaller time-step bias).
Estimator Functionality: The force-force estimator accurately provides fundamental frequencies and thermodynamic properties, while the displacement-displacement estimator precisely probes the lowest-energy phonon excitations (relevant for spectroscopy).
Anharmonicity Quantification: The simultaneous evaluation of both estimators provides a quantitative measure of the anharmonic strength within the material potential.
Real-World Application: Applied ab initio PIMD to diamond (benchmark) and the highly anharmonic I4₁/amd phase of atomic hydrogen at 500 GPa.
Key Finding (Hydrogen): Strong Nuclear Quantum Effects (NQE) and anharmonicity in I4₁/amd hydrogen cause a sizeable red-shift of approximately 250 cm^-1 in the vibrational spectrum.

Technical Specifications

Parameter	Value	Unit	Context
Simulation Framework	PIMD (TRPMD/PILD)	N/A	Uses PIOUD algorithm for integration.
Electronic Solver	DFT (PBE functional)	N/A	Used for computing ab initio Born-Oppenheimer forces.
Efficiency Gain (GEV vs Standard)	> 10	Factor	Overall gain in computational efficiency.
Hydrogen Phase Studied	I4₁/amd	N/A	Atomic hydrogen at 500 GPa.
Hydrogen Simulation Pressure	500	GPa	High-pressure regime.
Hydrogen Simulation Temperature	20, 120	K	Used to compare NQE and thermal effects.
PIMD Beads (Hydrogen, 20 K)	120	N/A	Number of beads used for convergence.
PIMD Time Step (Hydrogen)	0.9	fs	Integration time step used in PIMD.
Anharmonic Red-Shift (Hydrogen)	≈ 250	cm^-1	Shift in lowest-energy excitations due to NQE.
Diamond Simulation Temperature	300	K	Benchmark system.
Diamond Raman Shift (PIMD dxdx)	13.7 ± 2	cm^-1	Renormalization of the optical mode at Γ point.
Force Constant Matrix Definition	Kubo-transformed	N/A	Exact PIMD analogue of the classical force fluctuation relation.

Key Methodologies

The phonon dispersion curves are extracted from PIMD simulations using a multi-step procedure based on Kubo-transformed correlation functions and Generalized Eigenvalue Equations (GEV).

Path Integral Simulation: PIMD trajectories are generated using the Path Integral Langevin Dynamics (PILD) framework, integrated efficiently via the Fast Path Integral Ornstein-Uhlenbeck Dynamics (PIOUD) algorithm, ensuring constant temperature via a Langevin thermostat.
Force Calculation: Nuclear forces are computed ab initio at each time step using Density Functional Theory (DFT) (PBE functional) on the Born-Oppenheimer potential energy surface.
Correlation Matrix Generation: Real-space interatomic correlation matrices are built from the PIMD trajectories for two types of zero-time, Kubo-transformed quantum correlators:
- Force-Force (FF) correlators: &lt;&lt;Fi1 Fi2&gt;&gt;
- Displacement-Displacement (dxdx) correlators: &lt;&lt;δxi1 δxi2&gt;&gt;
Gauge Invariance: A pinning strategy is applied to the supercell atoms during correlation function accumulation to ensure the resulting matrices are fully symmetric and gauge invariant against global translational drifts.
Fourier Transformation: The real-space correlation matrices are Fourier transformed to obtain q-point dependent matrices, allowing access to the phonon dispersion across the Brillouin zone.
Generalized Eigenvalue Solution (GEV): The phonon frequencies are determined by solving the GEV problems (Eqs. 21 and 24), which significantly accelerates convergence compared to standard eigenvalue methods.
Dynamical Matrix Reconstruction: The dynamical matrix Dm(q) is reconstructed from the GEV eigenvalues and eigenvectors for each q-point.
Symmetrization and Interpolation: Space-group symmetry relations are imposed on the dynamical matrix elements. The final phonon dispersion is obtained by interpolating the fully symmetrized Dm(q) onto a finer q-grid.

Commercial Applications

This methodology provides crucial predictive capabilities for materials science and engineering, particularly in fields dominated by quantum effects and strong anharmonicity.

High-Temperature Superconductors: Essential for modeling the lattice dynamics and stability of high-pressure, hydrogen-rich superconductors (e.g., H₃S, LaH₁₀), where NQE and electron-phonon coupling are critical drivers of the superconducting state.
Thermal Management and Energy Storage: Accurate calculation of phonon dispersions and anharmonicity is necessary for predicting thermal conductivity, specific heat, and thermal expansion in advanced materials used in batteries and heat dissipation systems.
Spectroscopy and Sensing: The displacement-displacement estimator provides highly accurate predictions of low-lying phonon excitations, directly relevant for interpreting experimental Raman and Infrared (IR) spectroscopy data in quantum materials.
Computational Materials Discovery: The demonstrated 10x efficiency gain in PIMD calculations makes ab initio modeling of complex, strongly anharmonic systems feasible on modern supercomputing architectures, accelerating the search for novel functional materials.
High-Pressure Material Synthesis: Provides reliable theoretical data on the vibrational stability and phase transitions of materials under extreme pressures (hundreds of GPa), relevant for planetary science and industrial synthesis of ultra-hard materials (like diamond).

View Original Abstract

We devise an efficient scheme to determine vibrational properties from Path Integral Molecular Dynamics (PIMD) simulations. The method is based on zero-time Kubo-transformed correlation functions and captures the anharmonicity of the potential due to both temperature and quantum effects. Using analytical derivations and numerical calculations on toy-model potentials, we show that two different estimators built upon PIMD correlation functions fully characterize the phonon spectra and the anharmonicity strength. The first estimator is associated with the force-force quantum correlators and, in the weak anharmonic regime, yields reliable zero-point motion frequencies and thermodynamic properties of the quantum system. The second one is instead connected to displacement-displacement correlators and accurately probes the lowest-energy phonon excitations, regardless of the anharmonicity strength of the system. We also prove that the use of generalized eigenvalue equations, in place of the standard normal mode equations, leads to a significant speed-up in the PIMD phonon calculations, both in terms of faster convergence rate and smaller time step bias. Within this framework, using ab initio PIMD simulations, we compute phonon dispersions of diamond and of the high-pressure I41/amd phase of atomic hydrogen. We find that in the latter case, the anharmonicity is stronger than previously estimated and yields a sizeable red-shift in the vibrational spectrum of atomic hydrogen.

Tech Support

Original Source

DOI: https://doi.org/10.1063/5.0050450

References

1954 - Dynamical Theory of Crystal Lattices
1980 - Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra