In search of the electron-phonon contribution to total energy

1. Problem Statement

The total energy of a solid is a fundamental quantity governing material stability, phase transitions, and defect formation. Standard first-principles calculations (e.g., Density Functional Theory, DFT) rely on the Born-Oppenheimer (BO) approximation, which decouples nuclear and electronic degrees of freedom.

• Current Practice: The total energy is typically calculated as the sum of the static electronic energy (frozen nuclei) and the vibrational (phononic) energy (often treated via harmonic or quasi-harmonic approximations).
• The Gap: This two-step procedure is approximate. It neglects higher-order corrections arising from the coupling between electrons and phonons beyond the standard phonon contribution.
• Specific Confusion: Recent works (e.g., by P. B. Allen) suggested that summing the zero-point renormalization (ZPR) of electronic eigenvalues over occupied states constitutes an "electron-phonon contribution" to the total energy. However, it was unclear if this formulation was size-consistent and whether it represented a distinct physical contribution or a double-counting of existing phonon terms.
• Goal: The authors aim to rigorously derive the exact total energy expression within the BO framework, identify the specific terms constituting the electron-phonon contribution ($E_{elph}$), clarify the status of Allen's formula, and implement a practical method to compute these terms.

2. Methodology

The authors employ a rigorous perturbative expansion of the total Hamiltonian in powers of $\lambda = M_0^{-1/4}$, where $M_0$ is a characteristic nuclear mass.

• Exact Factorization & BO Expansion:
  • They start from the exact Schrödinger equation and decompose the total wavefunction into a product of a nuclear wavefunction and a BO electronic wavefunction, plus a correction term involving a self-energy operator ($\hat{\Sigma}$).
  • They derive an exact equation for the nuclear dynamics (Eq. 31) that includes a self-energy term condensing all electronic degrees of freedom.
• Mass-Scaling Analysis:
  • By scaling nuclear masses and displacements, they expand the total energy up to sixth order in $\lambda$.
  • They categorize terms based on their scaling:
    • $O(\lambda^0)$: Static BO electronic energy.
    • $O(\lambda^2)$: Harmonic phonon energy.
    • $O(\lambda^4)$: Anharmonic phonon-phonon terms, vector potential terms, and the lowest-order electron-phonon contribution.
    • $O(\lambda^6)$: Electronic inertial mass renormalization and higher-order mixed terms.
• Clarification of Allen's Formula:
  • They analyze the formula proposed by Allen (sum of ZPR of eigenvalues). Through mass-scaling analysis, they prove that Allen's term scales as $\lambda^2$ (second order), meaning it is actually a part of the phononic contribution (specifically the potential energy part of the zero-point energy), not a distinct fourth-order electron-phonon term.
• Implementation:
  • They derive a practical formula for the lowest-order electron-phonon contribution ($E_{elph}$) at fourth order, expressed in terms of electron-phonon matrix elements or derivatives of wavefunctions with respect to normal modes.
  • The implementation is carried out within the ABINIT software package, utilizing Density Functional Perturbation Theory (DFPT) and Fourier interpolation of perturbed potentials (including dynamical quadrupoles).

3. Key Contributions

1. Exact Formulation: Derivation of an exact expression for the total energy (Eq. 31) that includes contributions beyond the standard BO wavefunctions, explicitly identifying the self-energy operator.
2. Systematic Classification: A complete list of energy contributions up to sixth order in the mass-scaling expansion, distinguishing between:
   • Phononic contributions (harmonic, anharmonic, quasi-harmonic).
   • Electron-phonon contributions (arising from the Born term and self-energy).
   • Electronic inertial mass contributions (sixth order).
3. Resolution of the "Allen" Controversy:
   • Proved that Allen's proposed "electron-phonon contribution" is mathematically equivalent to half the zero-point energy of the phonons (specifically the potential energy part derived from electronic curvature).
   • Demonstrated that adding Allen's term to the total energy would constitute double-counting.
   • Confirmed the size consistency of the standard AHC (Allen-Heine-Cardona) band renormalization, refuting claims that it depends on unit cell size.
4. New Practical Formula: Derived and implemented the correct, size-consistent formula for the fourth-order electron-phonon contribution ($E_{elph}$), which scales as $M_0^{-1}$.

4. Results

The methodology was applied to two carbon allotropes: diamond (cubic) and lonsdaleite (hexagonal).

• Validation:
  • Calculations at the phonon zone-center ($\Gamma$) showed excellent agreement (< 0.5% difference) between DFPT and Finite Difference (frozen phonon) methods for phonon frequencies and energy terms.
  • Size consistency was verified by comparing primitive cells and supercells.
• Magnitude of Contributions:
  • Phonon Energy ($E_{ph}$): ~358 meV/2 atoms (dominant vibrational term).
  • Allen's Term ($E_{Allen}$): ~-104 meV/2 atoms. As proven, this is part of $E_{ph}$, not an additive correction.
  • Fourth-Order Electron-Phonon ($E_{elph}$): ~7.6 meV/2 atoms (approx. 3.8 meV/atom). This is the new, distinct contribution.
  • Electronic Inertial Mass ($E_{elm}$): ~-0.05 meV/2 atoms (negligible).
• Stability Analysis:
  • Diamond is more stable than lonsdaleite by 52.7 meV/2 atoms at 0 K when including all computed corrections.
  • While $E_{elph}$ is small, it is non-negligible for systems with shallow energy landscapes (e.g., magnetic configurations, defects, or competing polymorphs).
• Structural Effects:
  • Including phonon and electron-phonon forces leads to a zero-point lattice expansion of ~0.4% for diamond (increasing lattice parameter from 6.75 to 6.78 Bohr).
  • For lonsdaleite, the internal parameter $u$ is slightly reduced.

5. Significance

• Theoretical Clarity: The paper resolves long-standing ambiguities regarding the definition and calculation of electron-phonon contributions to total energy, correcting the misconception that eigenvalue renormalization sums constitute a separate energy term.
• Precision in Materials Design: By providing a rigorous, size-consistent method to compute the fourth-order $E_{elph}$, the work enables more accurate predictions of phase stability, particularly for materials where energy differences between polymorphs are on the order of a few meV/atom.
• Software Advancement: The implementation in ABINIT, including the handling of dynamical quadrupoles and non-linear core corrections, sets a new standard for high-precision total energy calculations in first-principles simulations.
• Future Applicability: The framework is essential for studying phenomena where quantum lattice effects are critical, such as superconductivity, magnetic phase transitions, and surface chemistry, where small energy differences dictate material behavior.