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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \omega _{\lambda }} of the following linear problem

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): A and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle A^*} describe the resonant and anti-resonant transitions between the occupied Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle v,v'} and unoccupied Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle c,c'} states

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): V and the screened potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): W .

The coupling between resonant and anti-resonant terms is described via terms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): B and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle B^*}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): B and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle B^*} . Hence, the TDA reduces the BSE to a Hermitian problem

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle AX_\lambda=\omega_\lambda X_\lambda~. }

In reciprocal space, the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): A is written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Omega is the cell volume, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \bar{V}} is the bare Coulomb potential without the long-range part

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \epsilon_\mathbf{G,G'}(\mathbf{q})} describes the screening in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): W within the random-phase approximation (RPA)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \omega _{\lambda }} and eigenvectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle X_\lambda} of the BSE

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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