Joint Approximate Diagonalization approach to Quasiparticle Self-Consistent $GW$ calculations

This paper introduces a Joint Approximate Diagonalization method for quasiparticle self-consistent $GW$ calculations that utilizes the full dynamical self-energy and a density matrix derived from the full Green's function, achieving accuracy comparable to standard $\mathrm{qs}GW$ while offering improved agreement with high-level CCSD(T) reference values.

The Gist

Imagine you are trying to tune a massive, complex orchestra (an atom or molecule) to play the perfect note. In the world of quantum physics, this "note" is the energy required to knock an electron out of the system, known as the Ionization Potential.

For decades, scientists have used a method called GW to predict these notes. However, the standard way of doing this is like trying to tune the orchestra by only listening to the first violin and assuming the rest of the instruments are perfectly in sync with it. This is the "single-shot" approach: you take a guess, calculate the note, and stop. If your initial guess (the "input") was slightly off, the final note will be wrong.

To fix this, scientists developed a "self-consistent" approach called qsGW. Think of this as a feedback loop: you play a note, listen to the result, adjust the tuning of the instruments, play again, and repeat until the sound is stable. However, the standard qsGW method has a shortcut. To make the math manageable, it forces the complex, changing sound of the orchestra into a simple, static, and symmetrical shape. It's like saying, "Let's pretend the orchestra only plays one perfect, unchanging chord," even though in reality, the sound is dynamic and messy.

The New Approach: "Joint Approximate Diagonalization" (JAD)

The authors of this paper, Ivan Duchemin and Xavier Blase, propose a new way to tune this orchestra. Instead of forcing the sound into a simple, static shape, they use a technique called Joint Approximate Diagonalization (JAD).

Here is the analogy:
Imagine you have a blurry, messy photograph of a crowd taken from a weird angle.

• The Old Way (Standard qsGW): You try to force the photo to look like a perfect, symmetrical grid. You erase the messy details to make it fit a simple rule.
• The New Way (JAD): Instead of forcing the photo to change, you rotate the camera (the mathematical "basis") until the messy crowd aligns as perfectly as possible. You don't erase the details; you just find the best angle where everyone lines up neatly.

In this new method, they look at the "Green's function" (which is like a map of all the possible energy states) at specific energy points. They rotate the mathematical "camera" until this map looks as diagonal (straight and clean) as possible.

The Key Difference:
The most important thing about this new method is that it doesn't throw away the messy, dynamic details. It keeps the full, complex, time-varying "self-energy" (the way electrons interact with each other) intact. It finds the best angle to view this complexity without simplifying it into a static, fake version.

The Results: Tuning the Orchestra

The authors tested this new method on a "test set" of 100 different molecules (the GW100 set).

1. Accuracy: Even though their new method is based on a completely different logic than the old standard method, the results were surprisingly similar. The difference in the predicted energy levels was tiny (about the size of a grain of sand compared to a mountain). This suggests both methods are finding the right "tuning," just by different routes.
2. The "Middle Ground" Improvement: They also tried a hybrid trick. In the standard method, they calculate the "density" (how many electrons are where) by just counting the occupied seats in the orchestra. But in the fully self-consistent method, they integrate the whole "sound wave" over time.
   • They created a new version called $\gamma$sGWJAD. This version calculates the electron density by integrating the full, complex wave (like listening to the whole concert) rather than just counting the seats.
   • The Result: This hybrid approach landed right in the middle between the standard method and the fully complex method. It turned out to be the most accurate of all, matching the "gold standard" reference calculations (CCSD(T)) even better than the others.

Summary

• The Problem: Standard methods for calculating electron energy either rely on bad starting guesses or simplify the complex physics too much.
• The Solution: A new method (JAD) that finds the best "viewing angle" for the complex data without simplifying the data itself.
• The Outcome: It works just as well as the current standard method but keeps the physics more realistic.
• The Bonus: By mixing this new method with a more thorough way of counting electrons, they created a "Goldilocks" scheme that is more accurate than both the standard and the fully complex methods, getting closer to the true experimental values.

In short, they found a way to tune the quantum orchestra by rotating the microphone to the perfect spot, rather than forcing the musicians to play a simpler song.

Technical Summary

Technical Summary: Joint Approximate Diagonalization Approach to Quasiparticle Self-Consistent GW Calculations

Problem Statement
The paper addresses the challenges inherent in ab initio many-body perturbation theory, specifically the $GW$ approximation for calculating quasiparticle energies. While the standard non-self-consistent $G_0W_0$ approach is efficient, its results depend heavily on the quality of the input mean-field orbitals (e.g., from DFT or Hartree-Fock). To mitigate this, self-consistent schemes are employed.

• Fully Self-Consistent $scGW$: Updates the Green's function ($G$) and screened Coulomb potential ($W$) self-consistently. While theoretically rigorous, it often yields ionization potentials (IPs) that deviate significantly from high-level reference methods like CCSD(T), typically underestimating IPs for small molecules.
• Quasiparticle Self-Consistent $qsGW$: A constrained scheme that updates quasiparticle energies and orbitals but relies on a static, symmetrized, and energy-independent ansatz for the self-energy ($\Sigma$). While this improves agreement with experiments compared to $G_0W_0$, it introduces an approximation that discards the full dynamical nature of the self-energy. Furthermore, $qsGW$ and $scGW$ often yield results differing by several tenths of an eV, with $qsGW$ tending to overestimate IPs relative to CCSD(T).

The authors identify a need for a quasiparticle self-consistent scheme that retains the full dynamical self-energy without resorting to a static symmetrized ansatz, while still providing a stable and accurate set of one-body orbitals.

Methodology
The authors propose a new scheme, termed $qsGW^{JAD}$, based on Joint Approximate Diagonalization (JAD) of the one-body GW Green's functions.

1. Core Concept: Instead of constructing an effective static Hamiltonian via a symmetrized self-energy (as in standard $qsGW$), the method seeks a unitary rotation ($U$) of the input Kohn-Sham molecular orbitals (MOs) that minimizes the off-diagonal elements of the set of Green's function matrices $\{G(\varepsilon^{QP}_n)\}$ evaluated at the input quasiparticle energies.

   • The objective function is: $\text{argmin}_U \sum_n || \text{off-diag}(U^\dagger G(\varepsilon^{QP}_n) U) ||$.
   • This approach preserves the full dynamical self-energy ($\Sigma_{GW}^{XC}(\omega)$) without approximation. The approximation lies solely in the fact that no single rotated basis can strictly diagonalize all $G(\varepsilon^{QP}_n)$ matrices simultaneously across the entire energy spectrum.

2. Self-Consistency Loop:

   • The optimized orbitals are used to update the density matrix and the non-interacting susceptibility ($\chi_0$).
   • A new dynamical self-energy is constructed.
   • Quasiparticle energies are extracted from the dominant poles of the spectral function $A_n(\omega)$ derived from the updated Green's function.
   • The JAD process is repeated to update the orbitals for the next iteration.

3. Intermediate Scheme ($\gamma sGW^{JAD}$):

   • The authors introduce a variant where the density matrix is not constructed simply by summing occupied quasiparticle orbitals (as in standard $qsGW$), but by integrating the full Green's function along the imaginary axis (as in $scGW$).
   • This captures both quasiparticle peaks and the incoherent background, creating a scheme intermediate between $qsGW$ and $scGW$.

Key Contributions

• Novel Formalism: Introduction of the JAD approach to $qsGW$, which bypasses the need for a static, symmetrized self-energy ansatz while maintaining a quasiparticle self-consistent framework.
• Dynamical Preservation: The method allows for the use of the full frequency-dependent self-energy, addressing a key limitation of standard $qsGW$.
• Hybrid Density Matrix: Demonstration that constructing the density matrix via imaginary-axis integration of the full Green's function yields an intermediate scheme ($\gamma sGW^{JAD}$) that bridges the gap between $qsGW$ and $scGW$.

Results
The methods were benchmarked on the GW100 molecular test set (reduced to 93 molecules to exclude those lacking all-electron basis sets), comparing against reference CCSD(T) data.

• $qsGW^{JAD}$ vs. Standard $qsGW$:

  • The new scheme yields results very close to the conventional $qsGW$ approach.
  • The Mean Absolute Error (MAE) between $qsGW^{JAD}$ and standard $qsGW$ is 65 meV (3 meV Mean Signed Error).
  • Both schemes show similar deviations from CCSD(T) (MAE $\approx$ 0.21–0.22 eV, MSE $\approx$ -0.15 eV), indicating that the JAD approach successfully replicates the accuracy of the standard static-ansatz method without the static approximation.

• $\gamma sGW^{JAD}$ vs. CCSD(T):

  • The intermediate scheme, which updates the density matrix via integration, shows improved agreement with CCSD(T) reference values.
  • The MAE reduces to 156 meV and the MSE to 62 meV.
  • This places the results between the standard $qsGW$ (which overestimates IPs) and $scGW$ (which underestimates IPs), moving closer to the CCSD(T) benchmark.

• Spectral Function Analysis:

  • In cases where the self-energy is strongly non-diagonal in the input basis (e.g., CO LUMO), the JAD rotation successfully produces a spectral function dominated by a single quasiparticle peak, similar to the outcome of enforcing diagonality in standard $qsGW$, but achieved through orbital optimization rather than self-energy approximation.

Significance and Claims
The paper claims that the $qsGW^{JAD}$ approach offers a robust alternative to the standard quasiparticle self-consistent scheme. Its primary significance is that it achieves comparable accuracy to the widely used $qsGW$ method while retaining the full dynamical nature of the self-energy, thereby avoiding the specific approximations inherent in the symmetrized static ansatz.

Furthermore, the authors demonstrate that the choice of density matrix construction is critical. By adopting a density matrix derived from the full Green's function (the $\gamma sGW^{JAD}$ variant), the method can be tuned to yield results that are closer to CCSD(T) reference values than either standard $qsGW$ or $scGW$. The authors suggest this intermediate scheme provides a promising route for improving the accuracy of quasiparticle calculations for molecular systems without incurring the full computational cost or accuracy penalties of fully self-consistent $scGW$.