GW+BSE:

• Excited states in the framework of many-body Green's function comprise charged excitations, where the number of electrons in the system changes from $N$ to $N-1$ or $N + 1$, and natural excitations, where the number of electrons remains constant.

• In the $|Nrangle rightarrow |N-1rangle$ case, an electron in the valence band (occupied orbital) is kicked out of the system by photon irradiation. In the $|Nrangle rightarrow |N+1rangle$ case, an electron from infinity falls into the conduction band (unoccupied orbital), emitting a photon simultaneously. These two processes are related to photoemission spectroscopy and inverse photon spectroscopy, through which we can study the electronic structure, ionization potential, and electron affinity of materials and molecules.

• In the $|Nrangle rightarrow |Nrangle$ case, an electron in the valence band is boosted into the conduction band after absorbing a photon, leaving a hole in the valence band. The excited electron and the hole left in the valence band are coupled together by Coulomb interaction, forming an exciton. The energy and oscillator strength of the exciton can be measured through optical absorption spectroscopy.

• Single-particle Green's function describes the electron addition or removal process in the system. If $|N,0rangle$ stands for the ground state of the $N$-electron system, then the single-particle Green's function is defined as: $$G(1,2) equiv G(vec{r}_1t_1,vec{r}_2t_2)=-ilangle N,0|T[hat{psi}(vec{r}_1t_1)hat{psi}^dagger(vec{r}_2t_2)]|N,0rangle$$ where $hat{psi}^{dagger}(vec{r}t)$ and $hat{psi}(vec{r}t)$ are the fermion creation and annihilation operators in the Heisenberg picture, respectively, $T$ is the Wick's time-ordering operator which has the effect of ordering the operators with largest time on the left. In the Lehmann representation, the solution for single-particle Green's function can be simplfied as the following quasiparticle Kohn-Sham-like equation: $$left[ -dfrac{1}{2}nabla^2+V_H(vec{r})+V_{ext}+Sigma[E_i^{QP}] right]psi_i^{QP}(vec{r})=E_i^{QP}psi_i^{QP} tag{1}$$ in which the self-energy $Sigma$ play the same role as the exchang-correlation functional in Kohn-Sham equation. However, solutions of Eq.(1) are quasiparticle energies and quasiparticle wavefunctions which are physically more meaningful than the solutions of Kohn-Sham equation.

• The motion of two-particle Green's function obeys the Bethe-Salpeter equation (BSE): $$L(1,2;1',2')=G(1,2')G(2',1')+int G(1,3)G(3',1')K(3,4';3',4)L(4,2;4',2')d(3,3',4',4)$$ where $L$ is the two-particle correlation function defined as: $$L(1,2;1',2')=-G_2(1,2;1',2')+G(1,1')G(2,2')$$ and $K$ is the two-particle (electron-hole) interaction kernel. The BSE can be turned into an eigenvalue problem: $$(E_c-E_v)A_{vc}^S+sum_{v'c'}K_{vc,v'c'}^{AA}(Omega_S)A_{v'c'}^S=Omega_SA_{vc}^S$$ in which $A_{vc}^S$ the exciton wavefunction and $Omega_S$ is the exciton egienvalue. By solving BSE eigenequation, the optical spectrum with electron-hole interaction can be obtained.

• Ref: GW method and Bethe-Salpeter equation for calculating electronic excitations: GW method and Bethe-Salpeter equation