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
The 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.
The 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.
The macroscopic dielectric which account for the excitonic effects is found via eigenvalues and eigenvectors of the BSE
How to
References
Pages in category "Bethe-Salpeter equations"
The following 23 pages are in this category, out of 23 total.