Category:Bethe-Salpeter equations: Difference between revisions

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The Bethe-Salpeter equation (BSE) is the Dyson equation for the four-point polarizability in the MBPT, which explicitly accounts for the electron-hole interaction. The BSE provides the state of the art approach for calculating the absorption spectra in solids.  
The Bethe-Salpeter equation (BSE) is the Dyson equation for the two-particle Green's function in the MBPT, which explicitly accounts for the electron-hole interaction. The BSE provides the state of the art approach for calculating the absorption spectra in solids.  
== Theory ==
== Theory ==
=== BSE ===
=== BSE ===

Revision as of 14:36, 16 October 2023

The Bethe-Salpeter equation (BSE) is the Dyson equation for the two-particle Green's function in the MBPT, which explicitly accounts for the electron-hole interaction. The BSE provides the state of the art approach for calculating the absorption spectra in solids.

Theory

BSE

The formalism of the Bethe-Salpeter equation (BSE) allows us to include the electron-hole interaction, i.e., the excitonic effects, in the calculation of the dielectric function. 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.

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.

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

How to

References