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 ${\displaystyle G_0W_0}$ 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

${\displaystyle A_{vc\mathbf{k}}^{v'c'\mathbf{k}'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'}\delta_{\mathbf{kk}'}+ \frac{2}{\Omega}\sum_{\mathbf{G}\neq0}\bar{V}_\mathbf{G}(\mathbf{q})\langle c\mathbf{k}|e^{i\mathbf{Gr}}|v\mathbf{k}\rangle\langle v'\mathbf{k}'|e^{-i\mathbf{Gr}}|c'\mathbf{k}'\rangle -\frac{2}{\Omega}\sum_{\mathbf{G,G}'}W_{\mathbf{G,G}'}(\mathbf{q},\omega)\delta_{\mathbf{q,k-k}'} \langle c\mathbf{k}|e^{i(\mathbf{q+G})}|c'\mathbf{k}'\rangle \langle v'\mathbf{k}'|e^{-i(\mathbf{q+G})}|v\mathbf{k}\rangle, }$

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