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 [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.

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., [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]

Scaling

The scaling of the BSE equation strongly limits its application for large systems. The main limiting factor is the diagonalization of the BSE/TDHF Hamiltonian. The rank of the Hamiltonian is

[math]\displaystyle{ N_{\rm rank} = N_k\times N_c\times N_v }[/math],

where [math]\displaystyle{ N_k }[/math] is the number of k-points in the Brillouin zone and [math]\displaystyle{ N_c }[/math] and [math]\displaystyle{ N_v }[/math] are the number of conduction and valence bands, respectively. The diagonalization of the matrix scales cubically with the matrix rank, i.e., [math]\displaystyle{ N_{\rm rank}^3 }[/math]. 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.,

[math]\displaystyle{ N_k\times N_q\times (N_v\times N_v\times N_G\times N_c\times N_c) }[/math],

where [math]\displaystyle{ N_q }[/math] is the number of q-points and [math]\displaystyle{ N_G }[/math] number of G-vectors.

How to

• BSE calculations

References