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 ${\displaystyle v,v'}$ 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

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

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

${\displaystyle N_{\rm {rank}}=N_{k}\times N_{c}\times N_{v}}$,

where ${\displaystyle N_{k}}$ is the number of k-points in the Brillouin zone and ${\displaystyle N_{c}}$ and ${\displaystyle N_{v}}$ are the number of conduction and valence bands, respectively. The diagonalization of the matrix scales cubically with the matrix rank, i.e., ${\displaystyle N_{\rm {rank}}^{3}}$.

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.,

${\displaystyle N_{k}\times N_{q}\times (N_{v}\times N_{v}\times N_{G}\times N_{c}\times N_{c})}$,

where ${\displaystyle N_{q}}$ is the number of q-points and ${\displaystyle N_{G}}$ number of G-vectors.

Solution of the BSE

Diagonalization

Time-evolution

How to

• BSE calculations

References