Category:Bethe-Salpeter equations: Difference between revisions

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where <math>N_q</math> is the number of q-points and <math>N_G</math> number of G-vectors.
where <math>N_q</math> is the number of q-points and <math>N_G</math> number of G-vectors.


== Solution of BSE ==
== Solving BSE ==
=== Diagonalization ===
=== Diagonalization ===
The exact diagonalization of the BSE Hamiltonian can be perform using various eigensolvers provided in ScaLAPACK, ELPA, and cuSolver libraries. The advantage of this approach is that the eigenvectors can be directly obtained and used for the analysis of the excitons.
The exact diagonalization of the BSE Hamiltonian can be perform using various eigensolvers provided in ScaLAPACK, ELPA, and cuSolver libraries. The advantage of this approach is that the eigenvectors can be directly obtained and used for the analysis of the excitons.

Revision as of 16:41, 16 October 2023

The formalism of the Bethe-Salpeter equation (BSE) allows for calculating the polarizability with the electron-hole interaction and constitutes the state of the art for calculating absorption spectra in solids.

Theory

Bethe-Salpeter equation

In the BSE, the excitation energies correspond to the eigenvalues of the following linear problem


The matrices and describe the resonant and anti-resonant transitions between the occupied and unoccupied states

The energies and orbitals of these states are usually obtained in a calculation, but DFT and Hybrid functional calculations can be used as well. The electron-electron interaction and electron-hole interaction are described via the bare Coulomb and the screened potential .

The coupling between resonant and anti-resonant terms is described via terms and

Due to the presence of this coupling, the Bethe-Salpeter Hamiltonian is non-Hermitian.

Tamm-Dancoff approximation

A common approximation to the BSE is the Tamm-Dancoff approximation (TDA), which neglects the coupling between resonant and anti-resonant terms, i.e., and . Hence, the TDA reduces the BSE to a Hermitian problem

In reciprocal space, the matrix is written as

where is the cell volume, is the bare Coulomb potential without the long-range part

and the screened Coulomb potential

Here, the dielectric function describes the screening in within the random-phase approximation (RPA)

Although the dielectric function is frequency-dependent, the static approximation is considered a standard for practical BSE calculations.

Macroscopic dielectric function

The macroscopic dielectric which accounts for the excitonic effects is found via eigenvalues and eigenvectors of the BSE

Scaling

The scaling of the BSE equation strongly limits its application for large systems. The main limiting factor is the diagonalization of the BSE Hamiltonian. The rank of the Hamiltonian is

,

where is the number of k-points in the Brillouin zone and and are the number of conduction and valence bands, respectively. The diagonalization of the matrix scales cubically with the matrix rank, i.e., .

Despite the fact that this matrix diagonalization is usually the bottleneck for bigger systems, the construction of the BSE Hamiltonian also scales unfavorably and can play a dominant role in big systems, i.e.,

,

where is the number of q-points and number of G-vectors.

Solving BSE

Diagonalization

The exact diagonalization of the BSE Hamiltonian can be perform using various eigensolvers provided in ScaLAPACK, ELPA, and cuSolver libraries. The advantage of this approach is that the eigenvectors can be directly obtained and used for the analysis of the excitons.

The following features are currently supported:

Time evolution

The alternative approach is to formulate the BSE as the initial-value problem for the macroscopic polarizability. This approach converges to the same solution as the exact diagonalization and can be used for obtaining the absorption spectrum, but does not yield the eigenvectors, which can be limiting for the analysis of the excitons. The advantage of this approach is the quadratic scaling with the size of the BSE Hamiltonian .

The following features are currently supported:

  • Obtaining the spectra
  • Calculations beyond Tamm-Dancoff approximation
  • Calculations of for

How to

References