DFTtoMLFT Energy definition for XPS calculation

Dear Quantum Developer,

Inspired by the XAS script for DFTtoMLFT, we aim to calculate the XPS.

Regarding the energy definitions, we encountered different scenarios. Let’s consider the following configurations:

1. $p^6L^{18}d^n$: ground state 2. $p^5L^{18}d^n$: excitation from the $p$-orbital to the vacuum 3. $p^6L^{17}d^{n+1}$: charge transfer (CT) from the ligand to the transition metal (TM)

Based on these scenarios, I derived the following equations:

$$6x + 18y + ndz + \frac{nd(nd-1)U_{dd}}{2} + 6ndU_{pd} = 0$$ $$5x + 18y + ndz + \frac{nd(nd-1)U_{dd}}{2} + 5ndU_{pd} = 0$$ $$6x + 17y + (nd+1)z + \frac{nd(nd+1)U_{dd}}{2} + 6(nd+1)U_{pd} = \Delta$$

Here, $ep_{\text{final}}$, $eL_{\text{final}}$, and $ed_{\text{final}}$ correspond to $x$, $y$, and $z$, respectively.

Are these equations correct?

After performing the calculations and plotting the XPS and XAS spectra on the same figure, I noticed a difference in the energy interval between the two peaks. The XPS peaks appear closer to each other compared to the XAS peaks, even though I used the same parameters for both calculations.

Could this difference be due to the coefficients in front of $U_{pd}$ in the above equations, or is there another explanation?

Best regards, Hamza