replacing the single-particle wavefunction $\phi_h$ of particle number $h$ by $\phi_k$. The index $\textbf{n}\left(h,k,l,m\right)$ means that two wavefunctions are changed correspondingly. The two-particle Coulomb matrix elements, $V_{hklm}$, are given by the following integral: $$\begin{aligned}
\label{Eq:2pME}
&V_{hklm}=\bra{\phi_{h}\phi_{k}}{\hat{C}}\ket{\phi_{l}\phi_{m}}\nonumber\\
&\equiv\frac{e^2}{4\pi\epsilon_0\epsilon_r}\int\int \frac{\phi_h\left(\textbf{r}_1\right)^*\phi_k\left(\textbf{r}_2\right)^*\phi_l\left(\textbf{r}_2\right)\phi_m\left(\textbf{r}_1\right)}{\abs{\textbf{r}_1-\textbf{r}_2}}\ d\textbf{r}_1\ d\textbf{r}_2.\end{aligned}$$ Depending on the order of the indices, these integrals include the direct Coulomb and the Coulomb exchange terms. For a symmetric harmonic confinement potential, analytic solutions for the Coulomb integrals can be found e.g. in Refs. .

The eigenfunctions of Eq. \[eq:H\] are obtained by diagonalizing $\hat{H}$ in the basis $\{\Psi_{\textbf{n}}\}$. The trion ground state has a small admixture of higher single-particle shells, which is the origin of the radiative Auger process. Upon optical recombination of one electron and a hole, the remaining electron of the trion is in a superposition including these higher shells. Detection of the frequency of the emitted photon projects the state of the remaining electron to the corresponding shell. For the trion, it is sufficient to carry out exact diagonalization for the initial state only since the final states are single-particle states.

In the dipole approximation, the emission spectrum can be computed with Fermi’s golden rule [@Chithrani2005]: $$I(\omega)=\sum_f\abs{\bra{\Psi_{\textbf{n}}^{(f)}}\hat{P}\ket{\Psi_{\textbf{n}}^{(i)}}}^2\cdot\delta(E_i-E_f-\hbar\omega),$$ where $\Psi_{\textbf{n}}^{(i)}$ is the initial state, $\Psi_{\textbf{n}}^{(f)}$ 