BSE+RPA Hybrid Schemes

• BSE+RPA hybrid schemes are computational methods that combine the Bethe–Salpeter equation for electron–hole interactions with the Random Phase Approximation for efficient screening of high-energy excitations.
• They restrict explicit BSE calculations to a small transition manifold near the gap while applying RPA corrections to high-energy states, thereby enhancing convergence and reducing computational cost.
• This approach accurately reproduces optical properties and energy-loss spectra by capturing both excitonic effects and plasmonic responses, closely matching experimental data.

BSE+RPA hybrid schemes refer to theoretical and computational approaches in many-body electronic structure theory that combine the Bethe–Salpeter equation (BSE) formalism for treating electron–hole correlations with the Random Phase Approximation (RPA) for treating collective screening effects and high-energy excitations. The motivation for these hybrid methods is to simultaneously retain the accuracy of BSE for bound exciton and low-energy optical responses while efficiently capturing high-energy dielectric screening and plasmonic responses via RPA, all while accelerating convergence and reducing computational scaling compared to pure BSE treatments.

1. Theoretical Foundations

The Bethe–Salpeter equation, originating from quantum field theory, provides a systematic framework for constructing the irreducible electron–hole (e–h) polarizability $\chi$ through a four-point Dyson equation: $$\chi^0(1,2;1',2') = \chi^0_{KS}(1,2;1',2') - \int d3d4\,\chi^0_{KS}(1,2;3,4)\,\tfrac12W(3,4)\,\chi^0(3,4;1',2')$$ where $\chi^0_{KS}$ is the noninteracting Kohn–Sham (KS) e–h propagator defined over a transition manifold $\mathcal T$, and $W$ is the statically screened Coulomb interaction. Direct solution of the full BSE is prohibitively expensive for realistic systems, as it involves diagonalization of a large (four-point) Hamiltonian. In contrast, the RPA sums an infinite series of non-interacting (bubble) diagrams, producing an efficient description of screening and high-energy (plasmonic) behavior but neglecting e–h attraction and vertex corrections critical for excitons.

The hybrid BSE+RPA approach, as embodied in the BSE+ (or "BSE-RPA hybrid") formalism, merges these methods by treating transitions near the gap using the full BSE and incorporating RPA-level corrections for all high-energy transitions outside the active manifold $\mathcal T$ (Søndersted et al., 2023, Olevano et al., 2018). This yields a corrected irreducible polarizability and a dielectric response that more faithfully matches experimental data, especially for energy-loss spectra and the real part of the macroscopic dielectric function $\epsilon_M(\omega)$.

2. Mathematical Formulation of BSE+RPA Hybrids

The BSE+ scheme replaces the conventional pure BSE dielectric response with an improved two-step construction:

1. Restricted BSE in $\mathcal T$: Solve the BSE within a small transition space ($\mathcal T$) containing only bands near the Fermi level, obtaining the four-point polarizability $\tilde P_{SS'}(q,\omega)$, which is then contracted to a two-point representation,

$$\tilde P_{GG'}(q,\omega) = \frac{1}{\Omega} \sum_{SS'} \rho_S(G) \tilde P_{SS'}(q,\omega) \rho_{S'}^*(G')$$

1. RPA Correction for High-Energy Transitions:
   • Compute the full KS-RPA polarizability $P^0(q,\omega)$ using all bands.
   • Define the irreducible polarizability,

   $$P^{\rm irr}_{GG'}(q,\omega) = \tilde P^{\rm irr}_{GG'}(q,\omega) - \tilde P^0_{GG'}(q,\omega) + P^0_{GG'}(q,\omega)$$

   where

   $$\tilde P^{\rm irr}_{SS'} = \tilde P^0_{SS'} - \tfrac12\sum_{S_1S_2}\tilde P^0_{SS_1}\,W_{S_1S_2}\,\tilde P^{\rm irr}_{S_2S'}.$$

• The improved two-point Dyson equation (BSE+):

  $$P^{\rm BSE+}_{GG'}(q,\omega) = P^{\rm irr}_{GG'}(q,\omega) + \sum_{G_1G_2}P^{\rm irr}_{GG_1}(q,\omega)V^{SR}_{G_1G_2}(q)P^{\rm BSE+}_{G_2G'}(q,\omega)$$

• The macroscopic dielectric function follows as $$\epsilon_M(\omega) = 1 - \lim_{q \to 0}\frac{4\pi}{q^2}P^{\rm BSE+}_{G=0,G'=0}(q,\omega)$$ (Søndersted et al., 2023).

This folding-in of high-energy transitions at the RPA level restores missing screening and guarantees convergence of the real part of $\epsilon_M(\omega)$ to experimental values with a minimal $\mathcal T$.

3. Alternative Hybridizations and Modern Variants

Other hybrid schemes interpolate between the GW+BSE kernel and self-consistent RPA (SCRPA) via frequency-dependent mixing,

$$K_{\rm hybrid}(\omega) = \alpha(\omega) K^{(0)}_{\rm EOM} + [1-\alpha(\omega)](v-W(0)) + K^{\rm dyn}_{\rm EOM}(\omega)$$

with a smooth function $\alpha(\omega)$, often chosen as $\alpha(\omega)=\Omega_p^2/(\omega^2+\Omega_p^2)$, $\Omega_p$ being the plasmon frequency. This recovers SCRPA at low energies (full ladder and vertex corrections) and pure GW+BSE screening at high energies (Olevano et al., 2018).

In the coupled-cluster (CC) diagrammatic framework, hybrid BSE+RPA schemes arise from decomposing the CC-BSE kernel into rings and ladder parts,

$K^{\rm hybrid} = K^{\rm RPA} + \Delta K^{\rm CC}$

where $K^{\rm RPA}$ is the usual ring (bubble) kernel and $\Delta K^{\rm CC}$ introduces ladder and exchange diagrams reflecting higher-order correlations. This construction leverages existing CCSD amplitudes to systematically upgrade the BSE kernel (Coveney et al., 2023).

Time-dependent stochastic BSE+RPA (TDsBSE) schemes recast the BSE as an effective time-dependent Schrödinger equation for quasiparticle orbitals, with direct Hartree and RPA-screened exchange kernels, evaluated using stochastic orbitals to achieve quadratic scaling (Rabani et al., 2015).

4. Computational Scaling and Efficiency

A central motivation for hybrid BSE+RPA methods is improved scaling and convergence. Pure BSE scales as $\mathcal O(N_H^3)$ with $N_H = N_k N_v N_c$ (k-points $\times$ valence bands $\times$ conduction bands), while pure RPA typically scales as $\mathcal O(N_k N_b^2 N_G)$, with $N_b$ the total number of bands and $N_G$ the number of $G$-vectors.

BSE+ minimizes computational cost by restricting explicit BSE solution to a small band manifold and reusing the full RPA polarizability, which is usually computed for screening in GW or BSE anyway. For high-energy screening, the cost of one RPA calculation is negligible compared to direct BSE diagonalization over a large band space (Søndersted et al., 2023). In practice:

• BSE+ achieves rapid convergence of $\mathrm{Re}\,\epsilon_M(\omega)$ with as few as 2–3 bands per side in $\mathcal T$, compared to hundreds for Kramers–Kronig convergence in BSE.
• Stochastic BSE+RPA approaches (TDsBSE) reduce the per-timestep cost to $O(N^2)$ by sampling occupied and screening subspaces with stochastic orbitals, as opposed to the $O(N^4-N^6)$ scaling of direct BSE eigenvalue methods (Rabani et al., 2015).
• Hybrid BSE+RPA-CC schemes retain the $O(N^6)$ scaling of CCSD, since $\Delta K^{\rm CC}$ is a correction evaluated from precomputed amplitudes (Coveney et al., 2023).

5. Practical Application and Validation

BSE+ and related hybrid schemes yield accurate low- and high-energy optical and loss spectra with substantial computational savings:

• Optical properties: BSE+ accurately reproduces refractive indices across a range of materials. For five prototypical solids, the mean absolute percent error in $n(\omega)$ is 2.6% (BSE+), compared to 15.9% (BSE) and 8.4% (RPA), using experimental benchmarks (Søndersted et al., 2023).
• Energy-loss spectra (EELS): BSE+ captures both low-energy excitonic structures and high-energy plasmon peaks, while BSE underestimates and RPA overestimates high-energy spectral weight.
• Large systems: Time-dependent stochastic BSE+RPA methods allow black-box evaluation of absorption spectra for systems up to $\sim$3000 electrons on standard hardware, enabling ab initio studies of nanocrystals and biological chromophores previously limited by BSE scaling (Rabani et al., 2015).

Practical recommendations for BSE+ implementation include using a PBE ground-state with a plane-wave cutoff $\sim 800$ eV, k-grids of at least 12–18 kpts/Å$^{-1}$, and energy window $\Delta E_B\sim 2$ eV for $\mathcal T$. Scissors shifts should be applied consistently across RPA, BSE, and BSE+.

6. Diagrammatic and Physical Interpretations

Hybrid BSE+RPA approaches retain distinct diagrammatic content compared to either method alone:

• Bubble diagrams (screening): RPA sums an infinite chain of bubble diagrams, representing collective screening and plasmons, but omits vertex corrections and bound excitons.
• Ladder diagrams (e–h attraction, vertex corrections): BSE and CC-based kernels include e–h ladder diagrams responsible for excitonic binding and satellite structures.
• Vertex mixing: In the EOM and CC formalisms, the hybrid kernel interpolates between the static-screened BSE and the full dynamical SCRPA vertex, avoiding double-counting and systematically improving correlation diagrams (Olevano et al., 2018, Coveney et al., 2023).

These diagrammatic distinctions clarify why BSE+RPA schemes simultaneously recover correct low-energy excitonic details and high-energy screening, and how their dynamical kernels capture plasmon satellites and spectral width more accurately than either BSE or RPA alone.

7. Limitations and Future Directions

While BSE+RPA hybrids systematically improve upon single-method approaches, several open challenges and directions remain:

• Frequency-dependent kernels: Accurate dynamical screening is crucial for high-energy features. Contour-deformation and Padé/plasmon-pole models have been recommended for capturing $K^{\rm dyn}(\omega)$ with manageable computational effort (Olevano et al., 2018).
• Self-Consistency and Vertex Corrections: SCRPA and CC-derived hybrids offer formal rigor, but full self-consistency and inclusion of higher excitations increase computational demands. Methods that efficiently compress basis sets (RI/Cholesky) and exploit block sparsity (Davidson/Lanczos) are important for scaling (Olevano et al., 2018, Coveney et al., 2023).
• Integration with GW and Beyond: Advancing BSE+RPA approaches often assumes GW-corrected quasiparticle energies; further integration with dynamical GW and time-dependent CC theories may yield even higher accuracy and consistency across neutral and charged excitations.

A plausible implication is that as stochastic methods, advanced vertex interpolations, and diagrammatic compressions mature, BSE+RPA hybrids will become the standard for optical and loss spectroscopy in complex solids and molecules.