Question

Why is it difficult to find eigenenergies and eigenstates of electrons in multi electron atoms? Why in the multi electron atoms do states with the same principal quantum number with different orbital quantum numbers have different energies? Why do states with the same principal quantum number and orbital quantum numbers have different energy levels depending on their total spin quantum number?

Answer 1

H^=−ℏ^2/2me(∇^2(1)+∇^2(2))+V(r1)+V(r2)+V(r12)

∇^2(1),∇^2(2),⋅,∇^2(11). The other big difference between single electron systems and multi-electron systems is the presence of the V(rij) terms which contain 1/rij, where rij is the distance between electrons i and j. These terms account for the electron-electron repulsion that we expect between like-charged particles.
Coulomb repulsion terms make it impossible to find an exact solution to the Schrödinger equation for many-electron atoms and molecules even if there are only two electrons. The most basic approximations to the exact solutions involve writing a multi-electron wavefunction as a simple product of single-electron wavefunctions, and obtaining the energy of the atom in the state described by that wavefunction as the sum of the energies of the one-electron components.

Electron in atom is described by 4 quantum no. : n, l, ml, and ms
yes energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n, are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and thus orbitals that belong to different subshells have different energies. Orbitals within the same subshell are still degenerate and have the same energy.
In hydrogen-line spectra, some lines are actually not single line but pairs of closely spaced lines. This is the called fine structure and this suggest that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations lead that electrons have a fourth quantum number,called spin quantum number(ms).
electron with ms=1/2 has a slightly lower energy in an external field in the positive z direction, and an electron with ms=−1/2 has a slightly higher energy in the same field.