Category:Bethe-Salpeter equations: Difference between revisions

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 ${\displaystyle {\omega }_{\lambda }}$ of the following linear problem

${\displaystyle \left(\begin{array}{cc}\mathbf{𝐀}& \mathbf{𝐁}\\ {\mathbf{𝐁}}^{*}& {\mathbf{𝐀}}^{*}\end{array}\right)\left(\begin{array}{l}{\mathbf{𝐗}}_{\lambda }\\ {\mathbf{𝐘}}_{\lambda }\end{array}\right)={\omega }_{\lambda }\left(\begin{array}{cc}\mathbf{𝟏}& \mathbf{𝟎}\\ \mathbf{𝟎}& -\mathbf{𝟏}\end{array}\right)\left(\begin{array}{l}{\mathbf{𝐗}}_{\lambda }\\ {\mathbf{𝐘}}_{\lambda }\end{array}\right).}$

The matrices ${\displaystyle A}$ and ${\displaystyle A^{*}}$ describe the resonant and anti-resonant transitions between the occupied ${\displaystyle v,v^{\prime }}$ and unoccupied ${\displaystyle c,c^{\prime }}$ states

${\displaystyle A_{vc}^{v^{\prime }{c}^{\prime }}=({\epsilon }_v-{\epsilon }_c){\delta }_{v{v}^{\prime }}{\delta }_{c{c}^{\prime }}+⟨c{v}^{\prime }|V|v{c}^{\prime }⟩-⟨c{v}^{\prime }|W|c^{\prime }v⟩.}$

The energies and orbitals of these states are usually obtained in a ${\displaystyle G_0{W}_0}$ 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 ${\displaystyle V}$ and the screened potential ${\displaystyle W}$.

The coupling between resonant and anti-resonant terms is described via terms ${\displaystyle B}$ and ${\displaystyle B^{*}}$

${\displaystyle B_{vc}^{v^{\prime }{c}^{\prime }}=⟨v{v}^{\prime }|V|c{c}^{\prime }⟩-⟨v{v}^{\prime }|W|c^{\prime }c⟩.}$

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., ${\displaystyle B}$ and ${\displaystyle B^{*}}$. Hence, the TDA reduces the BSE to a Hermitian problem

${\displaystyle A{X}_{\lambda }={\omega }_{\lambda }{X}_{\lambda }.}$

In reciprocal space, the matrix ${\displaystyle A}$ is written as

${\displaystyle A_{vc\mathbf{𝐤}}^{v^{\prime }{c}^{\prime }{\mathbf{𝐤}}^{\prime }}=({\epsilon }_v-{\epsilon }_c){\delta }_{v{v}^{\prime }}{\delta }_{c{c}^{\prime }}{\delta }_{{\mathbf{𝐤}\mathbf{𝐤}}^{\prime }}+\frac{2}{\mathrm{\Omega }}\sum _{\mathbf{𝐆}\ne 0}{\overline{V}}_{\mathbf{𝐆}}(\mathbf{𝐪})⟨c\mathbf{𝐤}|e^{i\mathbf{𝐆}\mathbf{𝐫}}|v\mathbf{𝐤}⟩⟨v^{\prime }{\mathbf{𝐤}}^{\prime }|e^{-i\mathbf{𝐆}\mathbf{𝐫}}|c^{\prime }{\mathbf{𝐤}}^{\prime }⟩-\frac{2}{\mathrm{\Omega }}\sum _{{\mathbf{𝐆},\mathbf{𝐆}}^{\prime }}{W}_{{\mathbf{𝐆},\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪},\omega ){\delta }_{{\mathbf{𝐪},\mathbf{𝐤}\mathbf{-}\mathbf{𝐤}}^{\prime }}⟨c\mathbf{𝐤}|e^{i(\mathbf{𝐪}\mathbf{+}\mathbf{𝐆})}|c^{\prime }{\mathbf{𝐤}}^{\prime }⟩⟨v^{\prime }{\mathbf{𝐤}}^{\prime }|e^{-i(\mathbf{𝐪}\mathbf{+}\mathbf{𝐆})}|v\mathbf{𝐤}⟩,}$

where ${\displaystyle \mathrm{\Omega }}$ is the cell volume, ${\displaystyle \overline{V}}$ is the bare Coulomb potential without the long-range part

${\displaystyle {\overline{V}}_{\mathbf{𝐆}}(\mathbf{𝐪})=\{\begin{array}{ll}0& \text{ if }G=0\\ V_{\mathbf{𝐆}}(\mathbf{𝐪})=\frac{4\pi }{|\mathbf{𝐪}+\mathbf{𝐆}{|}^2}& \text{ else }\end{array},}$

and the screened Coulomb potential ${\displaystyle W_{\mathbf{𝐆},{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪},\omega )=\frac{4\pi {ϵ}_{\mathbf{𝐆},{\mathbf{𝐆}}^{\prime }}^{-1}(\mathbf{𝐪},\omega )}{|\mathbf{𝐪}+\mathbf{𝐆}||\mathbf{𝐪}+{\mathbf{𝐆}}^{\prime }|}.}$

Here, the dielectric function ${\displaystyle {ϵ}_{\mathbf{𝐆},{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪})}$ describes the screening in ${\displaystyle W}$ within the random-phase approximation (RPA)

${\displaystyle {ϵ}_{\mathbf{𝐆},{\mathbf{𝐆}}^{\prime }}^{-1}(\mathbf{𝐪},\omega )={\delta }_{\mathbf{𝐆},{\mathbf{𝐆}}^{\prime }}+\frac{4\pi }{|\mathbf{𝐪}+\mathbf{𝐆}{|}^2}{\chi }_{\mathbf{𝐆},{\mathbf{𝐆}}^{\prime }}^{\mathrm{R}\mathrm{P}\mathrm{A}}(\mathbf{𝐪},\omega ).}$

Although the dielectric function is frequency-dependent, the static approximation ${\displaystyle W_{\mathbf{𝐆},{\mathbf{𝐆}}^{\prime }}(\mathbf{𝐪},\omega =0)}$ is considered a standard for practical BSE calculations.

The macroscopic dielectric which account for the excitonic effects is found via eigenvalues ${\displaystyle {\omega }_{\lambda }}$ and eigenvectors ${\displaystyle X_{\lambda }}$ of the BSE

${\displaystyle {ϵ}_M(\mathbf{𝐪},\omega )=1+\underset{\mathbf{𝐪}\to 0}{\lim }v(q)\sum _{\lambda }{|\sum _{c,v,k}⟨c\mathbf{𝐤}|e^{i\mathbf{𝐪}\mathbf{𝐫}}|v\mathbf{𝐤}⟩X_{\lambda }^{cv\mathbf{𝐤}}|}^2\times (\frac{1}{{\omega }_{\lambda }-\omega -i\delta }+\frac{1}{{\omega }_{\lambda }+\omega +i\delta }).}$

How to

• BSE calculations

References