BSE

Official Resources

• Homepage: https://etsf.polytechnique.fr/exc/about.html (EXC code)
• Documentation: https://etsf.polytechnique.fr/exc/
• License: Open-source (as per EXC code distribution)

Overview

BSE (Bethe-Salpeter Equation) solvers are a class of computational tools that solve the Bethe-Salpeter equation to calculate optical and dielectric properties of materials with excitonic effects. The BSE was first derived in particle physics (1951) and first applied to solids by Hanke and Sham, who calculated the absorption spectrum of bulk silicon. Since then, BSE has become the state-of-the-art method for ab initio simulation of absorption spectra.

Representative BSE solver codes include the EXC code from the European Theoretical Spectroscopy Facility (ETSF), which works in reciprocal space, frequency domain, and uses plane-wave basis. EXC calculates dielectric and optical properties including absorption, reflectivity, refraction index, and electron/X-ray energy loss for bulk systems, surfaces, and clusters. The BSE must be solved on top of: (1) a ground-state calculation, (2) a GW calculation for quasiparticle energies (or scissor operator approximation), and (3) an RPA screened Coulomb interaction calculation.

Scientific domain: Optical spectroscopy, excitonic effects, many-body perturbation theory
Target user community: Researchers studying optical properties of materials

Theoretical Methods

• Bethe-Salpeter equation for two-particle (electron-hole) correlation
• Screened Coulomb interaction W (from GW/RPA)
• Bare Coulomb interaction V (exchange term)
• Tamm-Dancoff approximation (TDA)
• Haydock iterative technique for large Hamiltonians
• Plane-wave basis for matrix elements
• Scissor operator approximation (alternative to GW)

Capabilities (CRITICAL)

• Optical absorption spectra with excitonic effects
• Dielectric function (real and imaginary parts)
• Reflectivity and refraction index
• Electron energy loss spectroscopy (EELS)
• X-ray energy loss
• Support for bulk, surfaces, and clusters (supercell method)
• Exciton energies and wavefunctions
• Multiple solver approaches (direct diagonalization, Haydock)

Inputs & Outputs

Input formats:

• Ground-state calculation output (wavefunctions, eigenvalues)
• GW quasiparticle energies (or scissor operator parameters)
• Screened Coulomb interaction (from RPA calculation)
• Plane-wave cutoff and transition space parameters

Output data types:

• Macroscopic dielectric function
• Absorption spectra
• EELS spectra
• Exciton energies and eigenvectors
• Oscillator strengths

Interfaces & Ecosystem

• Programming language: Fortran 90 (EXC code), with C and Perl utilities
• Linear algebra: LAPACK, BLAS, ATLAS, or vendor libraries (adaptive interface)
• FFT libraries: FFTW3, Goedecker-FFT, or vendor libraries
• Ground-state codes: Works with various DFT codes providing wavefunctions
• GW codes: Requires GW output for quasiparticle energies

Limitations & Known Constraints

• Requires preceding ground-state and GW calculations
• Hamiltonian size scales quadratically with transition space
• Memory-intensive for large systems
• Static W approximation typically used
• Computational cost limits system size

Performance Characteristics

• Direct diagonalization: O(N^3) for N transitions
• Haydock iterative: more efficient for spectral functions
• Parallelization over k-points and matrix elements
• Memory scales with number of G-vectors and transitions
• Multiple linear algebra library support for portability

Comparison with Other Codes

• vs VASP-BSE: VASP integrates BSE with DFT and GW; EXC is standalone
• vs Yambo: Yambo is a more comprehensive MBPT code; EXC focuses on BSE
• vs BerkeleyGW: BerkeleyGW includes BSE solver; EXC is dedicated BSE
• vs ABINIT-BSE: ABINIT integrates BSE in its workflow; EXC is external

Best Practices

• Ensure GW quasiparticle energies are well-converged before BSE
• Use scissor operator as cheaper alternative to full GW for simple systems
• Converge k-point sampling for exciton binding energies
• Use Haydock iterative method for large transition spaces
• Compare with and without TDA to assess approximation quality
• Test convergence with number of valence and conduction bands

Verification & Sources

Primary sources:

1. EXC code: https://etsf.polytechnique.fr/exc/about.html
2. G. Onida et al., Rev. Mod. Phys. 74, 601 (2002) - review of MBPT for optical spectra
3. Hanke and Sham, early BSE applications to solids

Confidence: VERIFIED - EXC code and methodology well-documented in literature