Revisiting ab-initio excited state forces from many-body Green's function formalism: approximations and benchmark

This paper presents a revised and practically implemented workflow for calculating ab-initio excited state forces using GW/BSE and DFPT formalisms, which corrects previous issues, improves accuracy through new approximations, and successfully applies the method to investigate exciton-phonon interactions and self-trapped excitons in systems ranging from the CO molecule to monolayer MoS$_2$.

The Gist

Imagine a material, like a piece of glass or a solar cell, as a giant, intricate dance floor. The atoms are the dancers, and the electrons are the music. Usually, the dancers move in a steady, predictable rhythm (this is the "ground state").

But what happens when you shine a bright light on them? Suddenly, a new, energetic dancer appears: an exciton. An exciton is a pair of dancers—a positive one and a negative one—holding hands and spinning wildly across the floor. This new energy changes the vibe of the whole room.

The problem is, we didn't really know how to predict exactly how the floor (the atoms) would react to this new, energetic couple. Would the floor warp? Would the dancers trip? Would the music change pitch?

This paper is like a new instruction manual for predicting how the dance floor reacts when these energetic couples show up. Here is the breakdown in simple terms:

1. The Old Map vs. The New GPS

For a long time, scientists had a map (a method called "finite differences") to guess how the floor would move. But this map was like trying to draw a coastline by taking a photo every time you moved a single step. It was slow, clunky, and often inaccurate for big, complex dances.

The authors of this paper revisited an old, slightly broken GPS system (a method from 2003) and fixed it. They created a new, high-speed GPS that can instantly tell you exactly how the atoms will push or pull when an exciton is present. They call this "Excited State Forces."

2. Fixing the "Ghost Push"

When the authors tried to use the old GPS, they found a weird glitch: the system sometimes told them the whole dance floor was being pushed in one direction, even though nothing was actually pushing it. It was like a ghost pushing the table.

They discovered the glitch wasn't in the physics, but in the math used to calculate the "footsteps" (electron-phonon coefficients). It was like a scale that was slightly uncalibrated. They fixed this by applying a "balance rule" (the Acoustic Sum Rule), ensuring that if the floor moves, it moves for a real reason, not because of a math error.

3. The "Renormalization" Trick

The old GPS used a low-resolution camera to see the dancers' movements. It missed the subtle, high-energy details. The authors realized they needed a high-definition lens.

They developed a clever trick called renormalization. Imagine you have a blurry photo of a runner. Instead of taking a new, expensive photo, you use a mathematical filter to sharpen the existing one based on how much faster the runner should be going. This allowed them to get high-precision results without needing a supercomputer the size of a city.

4. Testing the New System

To prove their new GPS works, they tested it on three different "dance floors":

• The CO Molecule: A tiny, simple duet. They checked if their math matched the "step-by-step" photos (the old method) and found it did, but much smoother.
• LiF (Lithium Fluoride): A crystal lattice. They showed how an exciton can get "stuck" in a hole it digs for itself (like a dancer tripping and digging a hole to hide in). This is called a Self-Trapped Exciton. Their method successfully predicted how the crystal would warp around this trapped dancer.
• MoS2 (Molybdenum Disulfide): A 2D material. They used their method to figure out which specific "vibrations" (phonons) the exciton likes to dance with. This helps explain why certain materials glow in specific colors when hit with light.

5. Why Does This Matter?

Why should you care if we know how atoms wiggle when excited?

• Better Solar Cells: Understanding these forces helps us design materials that don't break down when hit by sunlight.
• Faster Electronics: It helps us understand how energy moves through materials, which is crucial for faster computers.
• New Lasers and Sensors: By knowing exactly how light and matter interact, we can build better tools for medical imaging or communication.

The Bottom Line

Think of this paper as the difference between guessing how a trampoline will bounce by poking it with a stick, versus having a perfect mathematical model that predicts exactly how the fabric will ripple when a gymnast lands on it.

The authors didn't just build a better model; they fixed the broken parts of the old one, making it possible to simulate complex light-matter interactions quickly and accurately. This opens the door to designing new materials that can harness light in ways we've never seen before.

Technical Summary

1. Problem Statement

While ab initio techniques for calculating optical (GW/BSE) and vibrational (DFPT) properties of materials are well-established, the interaction between excitons and atomic vibrations (exciton-phonon coupling, or EPC) remains challenging to model efficiently.

• Limitations of Existing Methods: Traditional approaches often rely on "frozen phonon" (finite difference) calculations, which require $3N$ separate GW-BSE calculations for a system with $N$ atoms. Given that GW-BSE scales as $O(N^6)$, this is computationally prohibitive for all but the smallest systems.
• Issues with Previous Analytical Approaches: A seminal 2003 method by Ismail-Beigi and Louie (PRL 90, 076401) proposed an analytical way to calculate excited state forces (ESF) by combining BSE and DFPT. However, this method suffered from two critical issues:
  1. It produced non-zero net forces on the system's center of mass (violating Newton's third law) for molecules like CO.
  2. It relied on approximations that led to significant discrepancies (up to 4 eV/Å) compared to finite difference benchmarks.
• Gap: There is a lack of a practical, accurate, and efficient workflow to calculate ESF within the many-body Green's function formalism (GW-BSE) to study phenomena like self-trapped excitons (STE), polaronic excitons, and coherent phonon generation.

2. Methodology

The authors revisit the analytical theory of excited state forces, implementing a practical workflow that combines GW-BSE (using the BerkeleyGW code) and Density Functional Perturbation Theory (DFPT) (using Quantum Espresso).

Key Theoretical Formulations:

• Total Energy: The total energy of a system with an exciton concentration $x$ is defined as $E_T = E_{GS} + x\Omega_A$, where $E_{GS}$ is the ground state energy and $\Omega_A$ is the exciton energy.
• Excited State Forces (ESF): The force is defined as the negative gradient of the total energy with respect to atomic positions: $\vec{F}_T = \vec{F}_{GS} - x\nabla\Omega_A$.
• Hellmann-Feynman Theorem: Unlike the 2003 approach which expanded the basis before differentiation, the authors apply the derivative to the BSE Hamiltonian eigenvalue equation directly: $\nabla\Omega_A = \langle A | \nabla H_{BSE} | A \rangle$. This simplifies the expression and avoids unnecessary summations over intermediate states.
• Approximations:
  • Kernel Derivatives: The derivative of the electron-hole interaction kernel ($\nabla K_{eh}$) is assumed to be negligible. The authors validate this by showing kernel derivatives are a small fraction of the total excitation energy derivatives.
  • Renormalization of ELPH Coefficients: Standard DFPT calculates electron-phonon (ELPH) coefficients at the DFT level, which underestimates coupling compared to GW levels. The authors introduce a renormalization scheme (Eq. 5) to scale DFT off-diagonal ELPH coefficients using the ratio of Quasiparticle (QP) and DFT energy differences:
    $$\langle i|\partial H^{QP}|j\rangle \approx \left( \frac{E^{QP}_i - E^{QP}_j}{E^{DFT}_i - E^{DFT}_j} \right) \langle i|\partial H^{DFT}|j\rangle$$
  • Acoustic Sum Rule (ASR): To fix the non-zero center-of-mass force issue, the authors apply an ASR correction to the ELPH coefficients derived from DFPT, ensuring that pure translational modes yield zero force.

Implementation:

• A Python-based workflow connects Quantum Espresso (for DFPT/ELPH) and BerkeleyGW (for GW/BSE).
• The code handles complex phase consistency between wavefunctions and supports parallel computation.

3. Key Contributions

1. Resolution of the Center-of-Mass Force Issue: The authors identified that the non-zero net force in previous works was not due to the $\delta W/\delta G \approx 0$ approximation (as previously thought), but rather a failure of the DFPT ELPH coefficients to satisfy the Acoustic Sum Rule. They implemented an ASR correction that restores physical consistency.
2. Improved Accuracy via Renormalization: By introducing a renormalization scheme for off-diagonal ELPH coefficients, the method significantly improves agreement with finite difference benchmarks, particularly for bond lengths where the energy gap changes rapidly.
3. Practical Workflow: They provide a streamlined, open-source implementation that allows for the calculation of ESF and subsequent geometry relaxation of excited states without the prohibitive cost of finite difference GW-BSE calculations.
4. Validation: The method is rigorously benchmarked against finite differences for CO, LiF, and MoS$_2$, demonstrating high accuracy and convergence.

4. Results and Benchmarks

• CO Molecule:

  • The method reproduces finite difference results with high precision (deviations < 0.1 eV/Å after ASR and renormalization).
  • It correctly predicts repulsive excited state forces for both singlet and triplet excitons, leading to an elongation of the CO bond length in the excited state (consistent with experimental data).
  • The renormalization scheme improves accuracy for bond lengths $R_{CO} < 1.1$ Å.

• LiF (Self-Trapped Excitons):

  • Applied to a rocksalt LiF structure, the method successfully relaxes the system from a symmetric high-energy state to a Self-Trapped Exciton (STE) configuration with $C_{3v}$ symmetry.
  • Polaronic Excitons: In a $4\times4\times4$ supercell, the method identified polaronic excitons where a hole localizes on a fluorine atom, causing surrounding Li atoms to move away and F atoms to move closer.
  • Spectroscopic Signatures: The relaxation leads to a splitting of the valence band at the $\Gamma$ point, resulting in a redshifted absorption peak (Stokes shift $\approx 0.4$ eV) and a Zero Phonon Line (ZPL) shift.

• Monolayer MoS$_2$ (Symmetry Analysis):

  • The authors projected ESF onto phonon displacement bases to analyze exciton-phonon symmetries.
  • Group Theory Validation: The results confirm that the A exciton couples primarily to the $A'_1$ phonon mode, while the C exciton couples to both $A'_1$ and $E'$ modes. This aligns with experimental resonant Raman and transient absorption data.
  • Spin-Orbit Coupling (SOC): The study shows that SOC effects on ESF are negligible for MoS$_2$ when treated perturbatively, as the correction to the forces is small compared to the diagonal ELPH coefficients.

5. Significance and Impact

This work provides a robust, computationally efficient tool for studying exciton-phonon interactions in materials, bridging the gap between high-accuracy many-body theory and structural relaxation.

• Applications: The method enables the study of:
  • Self-Trapped Excitons (STE): Crucial for understanding broadband emission in perovskites and luminescence in wide-bandgap insulators.
  • Polaronic Excitons: Relevant for charge transport and optical properties in 2D materials.
  • Coherent Phonon Generation: Essential for interpreting ultrafast spectroscopy (e.g., Resonant Raman, pump-probe experiments).
  • Excitonic Insulators: Understanding lattice instabilities driven by excitonic condensation.
• Efficiency: By avoiding the $O(N^6)$ scaling of finite difference GW-BSE relaxations, this approach makes the study of excited state geometry optimization feasible for larger systems and supercells.
• Theoretical Rigor: The paper clarifies long-standing approximations in the field (specifically regarding kernel derivatives and ELPH renormalization), setting a new standard for accuracy in ab initio excited state dynamics.

In summary, Del Grande and Strubbe have successfully revitalized the analytical excited state force formalism, fixing historical numerical artifacts and providing a practical, high-accuracy framework for exploring the interplay between electronic excitations and lattice dynamics in complex materials.