Category:Bethe-Salpeter equations

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 ${\displaystyle v,v'}$ and unoccupied ${\displaystyle c,c'}$ states

${\displaystyle A_{vc}^{v'c'}=(\varepsilon _{v}-\varepsilon _{c})\delta _{vv'}\delta _{cc'}+\langle cv'|V|vc'\rangle -\langle cv'|W|c'v\rangle .}$

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 ${\displaystyle B^{*}}$

${\displaystyle B_{vc}^{v'c'}=\langle vv'|V|cc'\rangle -\langle vv'|W|c'c\rangle .}$

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

${\displaystyle AX_{\lambda }=\omega _{\lambda }X_{\lambda }~.}$

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

${\displaystyle A_{vc\mathbf {k} }^{v'c'\mathbf {k} '}=(\varepsilon _{v}-\varepsilon _{c})\delta _{vv'}\delta _{cc'}\delta _{\mathbf {kk} '}+{\frac {2}{\Omega }}\sum _{\mathbf {G} \neq 0}{\bar {V}}_{\mathbf {G} }(\mathbf {q} )\langle c\mathbf {k} |e^{i\mathbf {Gr} }|v\mathbf {k} \rangle \langle v'\mathbf {k} '|e^{-i\mathbf {Gr} }|c'\mathbf {k} '\rangle -{\frac {2}{\Omega }}\sum _{\mathbf {G,G} '}W_{\mathbf {G,G} '}(\mathbf {q} ,\omega )\delta _{\mathbf {q,k-k} '}\langle c\mathbf {k} |e^{i(\mathbf {q+G} )}|c'\mathbf {k} '\rangle \langle v'\mathbf {k} '|e^{-i(\mathbf {q+G} )}|v\mathbf {k} \rangle ,}$

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

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

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

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

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

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

${\displaystyle \epsilon _{M}(\mathbf {q} ,\omega )=1+\lim _{\mathbf {q} \rightarrow 0}v(q)\sum _{\lambda }\left|\sum _{c,v,k}\langle c\mathbf {k} |e^{i\mathbf {qr} }|v\mathbf {k} \rangle X_{\lambda }^{cv\mathbf {k} }\right|^{2}\times \left({\frac {1}{\omega _{\lambda }-\omega -i\delta }}+{\frac {1}{\omega _{\lambda }+\omega +i\delta }}\right)~.}$

How to

• BSE calculations

References