Category:Bethe-Salpeter equations

The formalism of the Bethe-Salpeter equation (BSE) allows for accounting the electron-hole interaction in the polarizability, which make the BSE the state of the art approach for calculating the absorption spectra in solids.

Theory

BSE

In the BSE, the excitation energies correspond to the eigenvalues [math]\displaystyle{ \omega_\lambda }[/math] of the following linear problem

[math]\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)~. }[/math]

The matrices [math]\displaystyle{ A }[/math] and [math]\displaystyle{ A^* }[/math] describe the resonant and anti-resonant transitions between the occupied [math]\displaystyle{ v,v' }[/math] and unoccupied [math]\displaystyle{ c,c' }[/math] states

[math]\displaystyle{ A_{vc}^{v'c'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'} + \langle cv'|V|vc'\rangle - \langle cv'|W|c'v\rangle. }[/math]

The energies and orbitals of these states are usually obtained in a [math]\displaystyle{ G_0W_0 }[/math] 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 [math]\displaystyle{ V }[/math] and the screened potential [math]\displaystyle{ W }[/math].

The coupling between resonant and anti-resonant terms is described via terms [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B^* }[/math]

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

Due to the presence of this coupling, the Bethe-Salpeter Hamiltonian is non-Hermitian.

A common approximation to the BSE is the Tamm-Dancoff approximation (TDA), which neglects the coupling between resonant and anti-resonant terms, i.e., [math]\displaystyle{ B }[/math] and [math]\displaystyle{ B^* }[/math]. Hence, the TDA reduces the BSE to a Hermitian problem

[math]\displaystyle{ AX_\lambda=\omega_\lambda X_\lambda~. }[/math]

In reciprocal space, the matrix [math]\displaystyle{ A }[/math] is written as

[math]\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, }[/math]

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

[math]\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}~, }[/math]

and the screened Coulomb potential [math]\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|}. }[/math]

Here, the dielectric function [math]\displaystyle{ \epsilon_\mathbf{G,G'}(\mathbf{q}) }[/math] describes the screening in [math]\displaystyle{ W }[/math] within the random-phase approximation (RPA)

[math]\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). }[/math]

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

The macroscopic dielectric which account for the excitonic effects is found via eigenvalues [math]\displaystyle{ \omega_\lambda }[/math] and eigenvectors [math]\displaystyle{ X_\lambda }[/math] of the BSE

[math]\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)~. }[/math]

How to

• BSE calculations

References