Bethe-Salpeter equation

The Bethe-Salpeter equation (BSE) was first derived and applied in the context of particle physics and QED in 1951 . The first application of BSE in solids was done by Hanke and Sham , who calculated the absorption spectrum of bulk silicon in qualitative agreement with experiment. Since then numerous works have shown successful applications of BSE for describing optical properties of materials and BSE has become the state of the art for ab initio simulation of absorption spectra.

Bethe-Salpeter equation

The Bethe-Salpeter equation for four-point polarizability $L(1234)$

$L(1234)=L_0(1234)+\int \mathrm{d}5\mathrm{d}6\mathrm{d}7\mathrm{d}8 L_0(1256)\Xi(5678)L(7834).$

Here, the common notation is used $1\to{\mathbf{r}_1,t_1}$. The interaction kernel is defined as $\Xi=\delta\Sigma/\delta G$. In practical calculations the self-energy $\Sigma$ is approximated via the GW approximation and after neglecting the higher order terms, the interaction kernel reduces to

$\Xi(1234)=\delta(12)\delta(34)v(13)-\delta(14)\delta(23)W(12).$

Thus, the interaction in BSE is described by two terms, the bare Coulomb interaction $v$ and the screened attractive potential $W$. The screened potential is the same potential used in the GW approximation. However, the important difference is that in BSE the static approximation is usually used, i.e., $W(\omega=0)$. The dynamical effects in the screened potential are shown to cancel out to a large extent with the dynamical effects in the polarizability $L_0$.

For practical reasons, the Bethe-Salpeter equation is usually reformulated in the the transition space and solved as a non-hermitian eigenvalue problem, where the excitation energies correspond to the eigenvalues $\omega_\lambda$

$\left(\begin{array}{cc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^* & \mathbf{A}^* \end{array}\right)\left(\begin{array}{l} \mathbf{X}_\lambda \\ \mathbf{Y}_\lambda \end{array}\right)=\omega_\lambda\left(\begin{array}{cc} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \end{array}\right)\left(\begin{array}{l} \mathbf{X}_\lambda \\ \mathbf{Y}_\lambda \end{array}\right)~.$

The matrices $A$ and $A^*$ describe the resonant and anti-resonant transitions between the occupied $v,v'$ and unoccupied $c,c'$ states

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

The energies and orbitals of these states are usually obtained in a $G_0W_0$ 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 $V$ and the screened potential $W$.

The coupling between resonant and anti-resonant terms is described via terms $B$ and $B^*$

$B_{vc}^{v'c'} = \langle vv'|V|cc'\rangle - \langle vv'|W|c'c\rangle.$

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

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., $B$ and $B^*$. Hence, the TDA reduces the BSE to a Hermitian problem

$AX_\lambda=\omega_\lambda X_\lambda~.$

In reciprocal space, the matrix $A$ is written as

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

where $\Omega$ is the cell volume, $\bar{V}$ is the bare Coulomb potential without the long-range part

$\bar{V}_{\mathbf{G}}(\mathbf{q})=\begin{cases} 0 & \text { if } G=0 \\ V_{\mathbf{G}}(\mathbf{q})=\frac{4 \pi}{|\mathbf{q}+\mathbf{G}|^2} & \text { else } \end{cases}~,$

and the screened Coulomb potential $W_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega)=\frac{4 \pi \epsilon_{\mathbf{G}, \mathbf{G}^{\prime}}^{-1}(\mathbf{q}, \omega)}{|\mathbf{q}+\mathbf{G}|\left|\mathbf{q}+\mathbf{G}^{\prime}\right|}.$

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

$\epsilon_{\mathbf{G}, \mathbf{G}^{\prime}}^{-1}(\mathbf{q}, \omega)=\delta_{\mathbf{G}, \mathbf{G}^{\prime}}+\frac{4 \pi}{|\mathbf{q}+\mathbf{G}|^2} \chi_{\mathbf{G}, \mathbf{G}^{\prime}}^{\mathrm{RPA}}(\mathbf{q}, \omega).$

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

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