Skip to content
6CCVD Research

Electronic structure and optical properties of quantum crystals from first principles calculations in the Born–Oppenheimer approximation

At a Glance

Section titled “At a Glance”
MetadataDetails
Publication Date2020-12-21
JournalThe Journal of Chemical Physics
AuthorsVitaly Gorelov, David M. Ceperley, Markus Holzmann, Carlo Pierleoni
InstitutionsUniversity of Illinois Urbana-Champaign, Centre National de la Recherche Scientifique
Citations10
AnalysisFull AI Review Included

Executive Summary

Section titled “Executive Summary”

This research develops and validates a highly accurate, non-perturbative computational framework for determining the electronic structure and optical properties of quantum crystals, focusing on systems where nuclear quantum and thermal motion significantly impact electronic behavior.

  • Methodological Breakthrough: A new formalism is established using Quantum Monte Carlo (QMC) methods to accurately account for the renormalization of electronic structure due to nuclear motion, moving beyond the limitations of the harmonic approximation and perturbative treatments.
  • Quantum Averaging (QA): The paper introduces a “Quantum Averaging” procedure for calculating optical conductivity (via Kubo-Greenwood formalism), which is essential for light nuclei (like hydrogen) at low temperatures where phonon quantization invalidates the traditional semi-classical (William-Lax, WL) approach.
  • Band Structure Restoration: A procedure is defined to meaningfully extend the concept of electronic crystal momentum (quasi-momentum) to quantum crystals by averaging the electron-ion wave function over the nuclear equilibrium distribution.
  • Hydrogen Gap Closure: Applied to C2/c-24 solid hydrogen (200 K, 250 GPa), the method confirms a large electronic gap reduction (on the order of 2 eV) primarily due to zero-point nuclear motion.
  • Optical Accuracy: QA successfully predicts the onset of optical absorption (Tauc gap) for hydrogen, yielding results consistent with the fundamental band gap, unlike the semi-classical WL method which significantly underestimates the gap near metallization pressures.
  • Validation: The methodology was also applied to carbon diamond (297 K), confirming that the differences between QA and WL methods are minimal for heavier elements, validating the necessity of QA specifically for light quantum systems.

Technical Specifications

Section titled “Technical Specifications”
ParameterValueUnitContext
Hydrogen Crystal StructureC2/c-24N/AHigh-pressure solid phase studied
Hydrogen Simulation Temperature200KCEIMC/RQMC calculations
Hydrogen Simulation Pressure Range248 to 290GPaRange of pressures studied for H2
Hydrogen Gap Reduction (ZPM)~2eVReduction in electronic gap due to nuclear motion
Hydrogen Tauc Gap (248 GPa, QA)1.3eVGap extracted via Quantum Averaging
Hydrogen Tauc Gap (248 GPa, WL)0.8eVGap extracted via Semi-classical Averaging
Carbon Diamond Temperature297KSimulation temperature for C
Carbon Diamond Supercell Size64AtomsCubic supercell used for PIMD
Carbon Tauc Gap (297 K, QA)3.55eVGap extracted via Quantum Averaging (PBE)
RQMC Imaginary Time Projection (t)2.00Ha-1Used for electron addition/removal energy convergence
RQMC Time Step (τ)0.01Ha-1Used for electron addition/removal energy calculation
QMC Twist Grid Size6x6x6 or 8x8x8N/AMonkhorst-Pack grid used for twist averaging

Key Methodologies

Section titled “Key Methodologies”

The methodology combines advanced many-body QMC techniques with path integral methods to handle both electronic correlation and nuclear quantum dynamics non-perturbatively under the Born-Oppenheimer (BO) approximation.

  1. Nuclear Dynamics Sampling:

    • Coupled Electron-Ion Monte Carlo (CEIMC) and Path Integral Molecular Dynamics (PIMD) were used to sample nuclear configurations (R) at finite temperature (T) and constant volume (V).
    • This sampling inherently includes nuclear zero-point motion and thermal fluctuations without relying on the harmonic approximation.

  2. Electronic Energy Calculation (Gap):

    • The fundamental electronic gap (Δ = µ+ - µ-) was determined from electron addition and removal energies using high-accuracy Reptation Quantum Monte Carlo (RQMC).
    • The Grand-Canonical Twist Averaging Boundary Conditions (GCTABC) approach was employed to minimize finite-size effects and calculate the electronic density of states (DOS) and energy density ε(µ).

  3. Band Structure Extension:

    • The electronic quasi-momentum (Bloch vector) for excitations in the quantum crystal was restored by marginalizing the total electron-ion wave function with respect to the nuclear equilibrium distribution.
    • This involved analyzing the overlap matrix elements T(q, m; q, m’) averaged over nuclear configurations, confirming that excitations retain a well-defined crystal momentum.

  4. Optical Conductivity Calculation:

    • Optical conductivity σ(ω, T) was computed using the Kubo-Greenwood (KG) formalism.
    • Quantum Averaging (QA): For light nuclei (H2), the standard semi-classical William-Lax (WL) procedure was replaced by QA, where the electronic eigenvalues and transition matrix elements are averaged over the nuclear states (Eq. 28), ensuring consistency with the thermodynamic gap definition at low temperatures.

  5. DFT Support:

    • Density Functional Theory (DFT) calculations (PBE, vdW-DF, HSE) were used for initial structural relaxation, generating trial wave functions for QMC, and performing comparative Tauc analysis on the absorption profiles.

Commercial Applications

Section titled “Commercial Applications”

This advanced computational framework provides critical predictive capabilities for materials operating under extreme conditions or where quantum effects dominate, enabling reliable design and engineering of next-generation materials.

  • High-Pressure Materials Discovery: Essential for accurately modeling materials (like hydrogen) under extreme pressures, guiding the search for high-temperature superconductors or novel metallic phases.
  • Advanced Semiconductor and Insulator Design: Provides highly accurate electronic band gaps and excitation spectra for wide-bandgap materials (e.g., diamond), crucial for high-power electronics and UV optics.
  • Predictive Optical Engineering: Enables precise prediction of optical absorption profiles and conductivity in quantum crystals, necessary for designing transparent materials or optimizing light-harvesting systems operating at cryogenic temperatures.
  • Quantum Materials Modeling: Offers a robust, non-perturbative tool for investigating strongly correlated systems and molecular crystals where electron-phonon coupling and nuclear quantum motion are significant, improving the reliability of ab initio simulations.
View Original Abstract

We develop a formalism to accurately account for the renormalization of the electronic structure due to quantum and thermal nuclear motions within the Born-Oppenheimer approximation. We focus on the fundamental energy gap obtained from electronic addition and removal energies from quantum Monte Carlo calculations in either the canonical or grand-canonical ensembles. The formalism applies as well to effective single electron theories such as those based on density functional theory. We show that the electronic (Bloch) crystal momentum can be restored by marginalizing the total electron-ion wave function with respect to the nuclear equilibrium distribution, and we describe an explicit procedure to establish the band structure of electronic excitations for quantum crystals within the Born-Oppenheimer approximation. Based on the Kubo-Greenwood equation, we discuss the effects of nuclear motion on optical conductivity. Our methodology applies to the low temperature regime where nuclear motion is quantized and, in general, differs from the semi-classical approximation. We apply our method to study the electronic structure of C2/c-24 crystalline hydrogen at 200 K and 250 GPa and discuss the optical absorption profile of hydrogen crystals at 200 K and carbon diamond at 297 K.

Tech Support

Section titled “Tech Support”

Original Source

Section titled “Original Source”

References

Section titled “References”
  1. 2010 - Momentum distribution and renormalization factor in sodium and the electron gas [Crossref]
  2. 2011 - Applications of quantum Monte Carlo methods in condensed systems [Crossref]
  3. 2013 - Quantum Monte Carlo applied to solids [Crossref]
  4. 2014 - Effect of electron correlation on the electronic structure and spin-lattice coupling of high-Tc cuprates: Quantum Monte Carlo calculations [Crossref]
  5. 2016 - Discovering correlated fermions using quantum Monte Carlo [Crossref]
  6. 2018 - Quantum Monte Carlo calculations of energy gaps from first principles [Crossref]
  7. 2020 - Direct observation of the momentum distribution and renormalization factor in lithium [Crossref]
  8. 2020 - Quantum Monte Carlo Compton profiles of solid and liquid lithium [Crossref]
  9. 2018 - Ab initio electronic structure calculations by auxiliary-field quantum Monte Carlo

Reads Similar to This

Investigations of the Optoelectronic and Structural Properties of Semiconductor Nanocrystals Under Perturbative Environments Impacts of channel constriction dimensions of graphene single-carrier transistors on the coulomb diamond characteristics Tunable Hybrid-Integrated Diode Laser at 637 nm