A Full Minimal Coupling GW-BSE Framework for Circular Dichroism in Solids: Applications to Chiral 2D Perovskites

Technical Summary: A Full Minimal Coupling GW-BSE Framework for Circular Dichroism in Solids

Problem Statement
Circular dichroism (CD) and chiroptical responses serve as critical probes for chirality and momentum-space geometry in solids. However, first-principles calculations of CD in periodic systems with strong excitonic effects, such as two-dimensional chiral hybrid organic-inorganic perovskites (2D-HOIPs), remain theoretically challenging. Traditional approaches rely on multipole expansions and sum-over-states (SOS) techniques. These methods suffer from significant limitations:

1. Gauge and Origin Ambiguities: Truncated SOS evaluations often yield results spuriously dependent on the choice of unit-cell origin or vector-potential gauge unless explicit corrections are applied.
2. Convergence Issues: SOS approaches converge slowly with the number of states, particularly in systems with dense band manifolds.
3. Numerical Instability: Standard velocity-gauge formulations exhibit divergences for intraband transitions or interband transitions between nearly degenerate bands.
4. Computational Complexity: Alternative length-gauge formulations (e.g., Blount's) require numerical derivatives of wavefunctions ($\nabla_k$), necessitating dense $k$-meshes and smooth gauges, which become computationally prohibitive for Bethe-Salpeter equation (BSE) calculations scaling as $O(N^5)$.
5. Incomplete Physics: Conventional Density Functional Theory (DFT) fails to capture many-body electron-hole correlations essential for accurate optical excitations in these materials.

Methodology
The authors introduce a Full Minimal Coupling (FMC) framework for calculating CD in solids, integrating the $GW$ approximation for quasiparticle energies with the Bethe-Salpeter equation (BSE) for electron-hole interactions.

• Theoretical Formulation: Unlike SOS methods that expand the light-matter interaction Hamiltonian in powers of the photon wavevector $q$ (retaining Electric Dipole (ED), Magnetic Dipole (MD), and Electric Quadrupole (EQ) terms separately), the FMC method retains the full minimal coupling interaction $p \to p + \frac{e}{c}A$ without expanding in $q$.
• Matrix Elements: The method evaluates optical matrix elements directly at a finite photon wavevector $q$. The excitonic matrix element is computed as $\langle 0| e^{iq\cdot r}p_x |S\rangle = \sum_{vck} A^S_{vck,q} \langle vk+q| e^{iq\cdot r}p_x |ck\rangle$.
• Key Advantages:
  • Gauge Invariance: The formulation is manifestly gauge-invariant, eliminating origin ambiguities.
  • Intraband and Near-Degenerate Transitions: By treating the perturbation to first order in $q$ (including spatial dispersion), FMC naturally includes intraband contributions and handles transitions between nearly degenerate bands without the energy denominator divergences plaguing SOS.
  • Equal Footing: ED, MD, and EQ contributions are treated on an equal footing without requiring separate, computationally intensive calculations of orbital angular momentum or quadrupole derivatives.
• Approximations: The method approximates the canonical momentum as $-i\hbar\nabla$ to avoid numerical complications from dual momentum shifts, thereby omitting non-local pseudopotential contributions (an effect explored in the Supplementary Information).
• Implementation: The framework is applied to two prototypical 2D-HOIPs: (S-NEA)$_2$PbBr$_4$ (S-NPB) and (S-MBA)$_2$PbI$_4$ (S-MPI). Ground states are calculated via DFT (GGA), quasiparticle energies via $G_0W_0$, and excitonic effects via BSE in the Tamm-Dancoff approximation.

Key Results

• Role of MD and EQ Channels: Calculations on S-NPB reveal that Magnetic Dipole (MD) and Electric Quadrupole (EQ) channels contribute equally in magnitude but with opposite signs to the CD signal. Their partial cancellation results in a significantly smaller total CD signal than predicted by MD-only models.
• Excitonic Effects and Lineshape: In S-NPB, the inclusion of excitons and the EQ term leads to an asymmetric peak structure in the CD spectrum, contrasting with the symmetric "Cotton effect" predicted by models neglecting EQ or molecular states. The convergence of the CD spectrum in SOS requires over 1,000 bands to capture the asymmetry introduced by molecular states, whereas FMC converges more robustly.
• Numerical Stability in Degenerate Systems: For S-MPI, which features a band structure with nearly degenerate states due to weak interlayer coupling, the SOS approach breaks down completely, producing oscillating and unstable spectra dependent on energy thresholds. In contrast, the FMC approach yields a qualitatively reliable and stable spectrum.
• Intraband Contributions: FMC explicitly captures sizable diagonal (intraband) transition matrix elements, which correspond to Berry connection-like terms. These terms are neglected in standard SOS formulations but are shown to be significant for the CD response.
• Orbital Angular Momentum: FMC provides smooth, physically consistent diagonal elements for orbital angular momentum matrices, whereas SOS results exhibit unstable fluctuations dependent on numerical parameters.

Significance and Claims
The paper claims that the FMC framework offers a computationally efficient and numerically robust alternative to conventional SOS and multipole expansion methods for predicting chiral optoelectronic phenomena in complex solids. By eliminating slow convergence issues, gauge ambiguities, and divergences associated with band degeneracies, FMC enables accurate first-principles predictions of CD in materials with strong excitonic effects and dense band structures. The study establishes that intraband and quasi-degenerate transitions, naturally captured by FMC, can dramatically reshape CD spectra, highlighting the importance of quantum geometry (Berry connection-like terms) in chiroptical physics. This work provides a foundation for understanding the origin of CD in chiral HOIPs and supports the development of chiral optoelectronic devices.