The energy of an electron on $H$ atom is given by the formula: $-13.6 \; \text{eV}/n^2$. The constant value is born from $H$ dielectric constant and efective mass of the electron.
My question is: this formula can be used to estimate the energy of a electron on a $Si$ Coulomb potential? Is it sufficient to user the proper values of $\epsilon$ and electron effective mass?
The answer is no. Hydrogen atom has only one electron. That's why it's possible to get the simple formula for energy. For other atoms, such as silicon, there are more than one electrons around the nucleus, and they interact with each other. In general it's very difficult to calculate a many-body system. A simple formula like the one for hydrogen is not possible. However, considering the shell structure, there's this concept of pseudopotential, which tries to approximate the effects of the nucleus and inner electrons on the outer electrons with a simplified potential. It's yet a quite complex method.
